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Collective effects in the jn configuration
N&ear Physics Af75 (1971) 272-288;
@
~o?tb-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint oc microfilm without written permission from the publisher
COLLECTIVE EFFECTS IN THE j” CONFIGURATION M. VALLIERES
t and R. M. DREIZLER
*t
Department of Physics, University of Pennsylvania, Phi~ade~hia, Pa. 19104, USA Received 10 May 1971 (Revised 2 July 1971) Abstract:
Using a model Hamiltonian with pairing and quadrupole forces we investigatethe structure of selectedstates of the cotiguration j” withj = Idp,3, T and n = 4,6 as a function of the reIativecouplingstrength. Exact results for excitation energies, E2 transition rates and twoparticle transfer spectroscopic amplitudes are presented and compared with the results of pre-
vious approximate variational calculations for the same model.
1. Introduction Collective effects in nuclei have been attributed to the dominance of the long-range components of the residual nuclear forces over the short-range part. Several simplified models have been suggested for the investigation of the underlying many-body aspects. Among these, the model using a single&shell and a pairing plus quadrupolequadrupole Hamiltonian offers a convenient starting point for more ambitious microscopic descriptions. This model has been used in particular as a testing ground for new approaches to the many-body problem. These methods are characterized by the direct calculation of physically observable quantities such as multipole matrix elements and spectroscopic amplitudes for particle transfer without the explicit in~rvention of wave functions. A main tool is the exploitation of sum rules for these matrix elements. So far, these efforts have concentrated on the emergence of rotational characteristics in the pure quadrupole limit 2- “) an d on the behavior of the properties of low-lying states in the transition from the pure pairing to the pure quadrupole interaction “). The present investigation has been motivated largely by the desire to compare the results of the simple scheme of ref. “) with exact solutions. These are available for four- ‘9“) (j = $-J$) and six- *) (1 = 9, -?i_) particle systems, but the main emphasis in these investigations was centered on energy spectra. With the emphasis on collective features, we are also interested in the transition amplitudes to states of the same and neighboring nuclei. In addition “collective effects” are more pronounced for larger j-values and particle numbers (that do not approach the mid-shell point N = _)(2j+ 1)). t Supported by a stipend from the Ministbre de I’Education, Quebec, Canada. tf Supported in part by the US Atomic Energy Commission. 272
COLLECTIVE EFFECTS
273
We have thus attempted to provide exact results for as large a system j” as was accessible with our computer facilities. This turned out to be the configuration (y)“. The procedure used is the standard shell-model technique. We calculate singleparticle coefficients of fractional parentage in the seniority basis using a modified version of a code provided by Bayman and Lande ‘). We then set up the Hamiltonian matrix and diagonal& it. Relevant details are contained in sect. 2, together with a description of the model and a brief resume of the variational method of ref. “). In sect. 3 we present the exact results and compare them with those obtained by the variational method. In all we place emphasis on the variation of the results with the relative strength of the force components, thej-value (y, 9, y) and particle number (N = 4,6). 2. Theory 2.1. THE MODEL
The model Hamiltonian contains a pairing part with the force constant G and a quadrupole part with the strength x. We write it in the form H = -G~A~+A~-~xC(-)~B,~B~_~, Q Q = j+*.
(1)
Single-particle terms are not relevant in this problem. The operators A:+, A: and Bi are special cases of the general two-particle transfer and multipole operators that
can be constructed from the single-shell fermion creation and destruction operators
B: = (2k+1)-*C
m
[A qjm
k-J (-)i-n’+%m+am_q.
These operators can be shown to span the group 0(4j+2) with the Lie algebra [g,
A$+] = ~2J2k’+l(-)2’-k”
[; ;: ,“;;,I(::
5’ ;“) Akq:q,,
k”
[A;, A;:+] =
a,,,6,,
- 2 c ( -)2i+q[(2k+
1)(2k’+ 1)(2k”+ I)]+ X [
[B;, B::] = g ( -)2j-k”[1
(4)
_;
;:
,r”l’,] (; :’ :“)Bfz:‘_, ,
(5)
_( _)k+k’+k”]
xJ2k”+l
[;
;: ,f”;‘,1(;
:’ y)B:;,,.
(6)
We have used the notation [ ] for Clebsch-Gordan coefficients and { } for 6-j symbols.
274
M. VALLIBRES AND R. M. DREIZLER
In the pure pairing limit (x = 0) we only have to consider the SU(2) subalgebra spanned by the operators A:+, Ai and the number operator N = (2j+ 1)*I$. This leads to the well-known seniority scheme lo). In the pure quadrupole limit (G = 0) we have a situation reminiscent of the SU(3) scheme ‘I). If we consider the commutation relations of the operators B,’ and Bt, we find
(8)
Eq. (7) identifies Bi as the angular momentum operator B: = [j(j+l)(2j+l)]+Jq;
(10)
however the identifi~tion of Bi as the SU(3) quad~pole tensor is spoiled by the appearance of the Bi term in the commutator (9). Arima 12) has shown that the typical rotational band structure of the SU(3) scheme is preserved for states IJMa), provided that J < 2j-t 1 and j B 1. 2.2. THE EXACT
SOLUTION
For the exact solution of the problem we first construct a set of basis states in terms. of the coefl?cient of fractional parentage (c.f.p.) in the seniority limit
The antisymmetric iv-particle state Y(jNsc&W), classified by seniority s, anguiar momentum I, M and a counting label CL,is expressed in terms of the angular momentum coupled set of (N- I)-particle and one-particle states. In the language of second quantization, the c.f,p. are defined by the relation
The matrix elements of the Hamiltonian operator (1) in the basis &V)sollM> can be evaluatedin terms of the c.f.p. connecting the IV-and (N- l)-particle systems and the c.f.p. connecting the (N- I)- and (N-2)-particle systems. The latter are defined as <(N-l)vBJ~la,tI(N-2)salM)
= x/N- 1
(131
275
COLLECTIVE EFFECT‘S
The resulting expression for the pairing and quadrupole interactions is ((N)saZMIA~+A~((N)s’a’ZM)
= ~3,~N(N-l)
4Q(2(21+1)
-N(N-1)
x Jvfl ,
J’v’p’. I”a”S”
x u~~~~~~~~~(Jv~)u I’,~,,ct”(J’v’s’)vJ,y,~,~Zs’a’).
(15)
Here the curly bracket stands for a 9-j symbol. The calculation of the c.f,p. in the seniority limit was performed with a code provided by Bayman and Lande ‘). The code uses the fact that the basis states I(N)s belong to the representation [l, F,r ‘,, I] of SU(2j+ 1) and transform irreducibly under Sp(2j+ 1). If one evaluates the Casimir operators (G) of both groups in the basis of non-antisymmetric product states PW
- ~)V~~~)~(~)l~,
and diagonaiizes the resulting matrices, the physical eigenstates can be identified by inspection of the eigenvalues. The c.f.p. are then given by the appropriate eigenvectors of the matrix that diagonalizes an arbitrary combination a(G(SU(2j+
1))) +b(G(Sp(2j+
1))).
A convenient check of the calculated c.f.p. is provided by eq. (14). The pairing part of the Hamiltonian is diagonal with respect to the basis (11) (in the counting label a enforced by orthogonalization). The main limitation of this method lies in the dimensionality of the (over complete) set of states [Y(N-- 1)4]5. In table la we give the dimensionality of the product basis TABLE la
The number of basis states for the case of six particles as a function of angular momentum J and
the shell label j
I=0
I=2 x=4
9
9
21
23
T
YE-
zs z
19( 8) 89(16) 153(26)
26(10) 147(23) 194(37)
36(13) 173(31) 302(S)
47(16) 230(41) 317(68)
61(20) 296(53) 409(88)
The number of physical states with symmetry (1, r. N. + . ., I] is given in brackets.
27
7
77( 24) 378( 67) 523(112)
276
M. VALLIBRES AND R. M. DREIZLER
together with the number of physical states with angular momentum 0,2,4 of selected six-particle problems. An alternative method for the calculation of the c.f.p. has been suggested by Hassitt 13). With the c.f.p. we can calculate the Hamiltonian matrix using eqs. (14) and (1.5). After diagonalization the eigenvectors are of the form
The individual matrix elements of the multipole and transfer operators Z$ and Ai f for the eigenstates (16) are obtained by the following- steps: We first calculate the reduced matrix elements in the seniority basis, using the definition ((N)ZMs,IB~I(N)Z’M’,‘cr’) = [2Z + l]-* ~(~)Z~~~l~~~l~~-
Ml
“, k ’
CM 4
B,((N)Zsa, (N)Z’s’a’),
2)Z’M’s’a’) = [(l --Sk, e)2Z$ I]-+ A,((N)Zsa, (N - 2)Z’s’a’).
The results are B,((N)ZSol, (N)Z’s’cL’)= N[(2z+ 1)(2i’+ 1)]3( --y-r
~~(~~)ZS~,(~-~~~'S'~') = (-)k""-"[&N(N- 1)(2k+1)(21+1)]
Next, the reduced matrix elements among the eigenstates are obtained by Z3&, I’?‘) = C C~y~~(Zsa, I’s’a’)C$f?, se,S’.’
A&y, I’y’) = 2.3. VARIATIONAL
c CE’yAk(Zsa, Z’s’a’)C~~72)r’y’. SC& S’OI’
METHOD
In this section we give a brief summary of the variational calculation proposed by Dreizler and Klein “). The ground state expectation value of the IIamiltonian is evaluated under the assumption that the sums over intermediate states occurring in the matrix elements (OOlA~+ -4~100) and (OOl’&( --r Bt B!, lO0) are saturated by only one state
<~ol~oo+4l~> = r: l&(O, WI2 = l4(O@), 4
qN4))l:=,
= c lB,(2y, 0)l” a IZ3,(2,0)1&~ * I
,
(21) 122)
COLLECTIVE EFFECTS
271
The gro~d-state-to-ground-state two-particfe transfer amplitude A ~~~~ and the quadrupofe transition matrix element to the first excited state of angular momentum two &(2,0) are calculated by a variational principle with the following kinematical restrictions (A,&, 1’) = A,(I(N--21, l’(N-4)):
The first two equations are obtained from the afgebra, eqs. (4) and (5) with k = fc’ = 2, under the same ~surn~io~s as for the eateulation of the ground state energy. The last equation is derived from the Casimir operator of the full group 0(4i+ 2) under the czdditroml assumption that correlations of multipole order greater than two can be neglected for its ground state expectation value, With the four amplitudes Ae(0, 0), A2(2, 0), A2(0, 2) and ,S,(2,0) obtained from the ground state variational principle, we can calculate additional ones f&(2,2), J&(2,2>, M2,2)3 using the sum rules (241 4(2, @4,(2* 2)-Q%<@ 2)4.(2,2} 4@, 2&<(2,
A,@,
O)B&
= 50 (5 ‘i ;) I&(2,
O),
(27)
q--B,f2,O)f%z(2,2) = 10 (5 ; ;) A&O, 21,
2j--B,<@
0~~2~22~
=
-
10
;
;
;
t
A2@,
0).
cw
I
These sum rules are also consequences of the 0(4j+2) algebraand the assumption of one state saturation in the intermediate state summation. In addition, one can obtain an approximate expression for the excitation energy OFthe first excited state with angular moments two
%o = from the commutator
E,--E,,
[.B&Hj. This yields
9 X+@ A*(O*0) B(2 a20 = Sj(j + 1)(2j + 1) 2 3
fA2@9
0)
-A&h
211.
P9
As stated in the introductions one of our objectives is the comparison of the results
278
M. VALLIeRE.3
AND R. M. DREIZLER
x
00
0
a=56a 2 4
(x/51
Fig. 1. The spectrum of the J = 0,2, 4 states of the configuration (+)’ as a function of the relative coupling strengths c( = x/SQG -+ 0 to 1 and p = 5GQ/x + 1 to 0. The energies are scaled to/RG on the left-hand side of the graph and to 5/x on the right-hand side.
-ix
x/5
Fig. 2. The same as fig. 1 for the configuration (J#)6. For clarity we have not included all the highly excited states.
279
COLLECTIVE EFFECTS
-ai
s=o
(‘7/d
-3.0 -
I
Fig. 3. The same as fig. 2 for the configuration (J$)4.
Fig. 4. The same as fig. 2 for the configuration (AJ)‘j.
280
M. VALL@RES
AND R. M. DREIZLER
2 4
x/5
-3.0 -
Fig. 5. The same as fig. 2 for the configuration
(9)‘.
x/5
I
Fig. 6. The same as fig. 2 for the configuration
(y)“.
COLLECTIVE
281
EFFECTS
of this variational calculation with the exact solutions. In addition, we would like to check the quality of the saturation of the various sum rules with one intermediate state. This, together with a discussion of the exact results in their own rights, will be presented in sect. 3. 3. Results 3.1. ENERGY
SPECTRA
We have performed the calculations outlined in sect. 2.2 for states with I = 0, 2, 4 in the configurations (y)4, (y)6, (J$)4, (y)” and (q)4, as well as for states with I = 0, 2 in the configuration (y)“. The spectra are shown in figs. l-6 as a function of the relative strength of the two parts of the interaction. The relative strength is TABLE lb The number of physical states of given angular momentum in each configuration
State
8 N=4
I=0
I=2 I==4
9
9 N=6
N=4
N=6
N=4
N=6
4 7 11
3 5 7
8 16 26
4 6 9
13 31 51
2 4 5
-
measured by a = x/5GQ on the pairing side and by j?l = 5GSZ/x on the quadrupole side. Note also that the energies are given in units of QG on the pairing side and in units of 3x on the quadrupole side. For clarity we have not included in these figures all the highly excited states. The total number of states for each angular momentum are listed in table lb. We ,remark the following: in the limit of a pure quadrupole force (/I = 0) we expect the lowest state of each angular momentum to form a rotational band. According to Arima “),the ratio of excitation energies R4 = (E4-Eo)/ (E, -E,) should approach the ideal value of q with increasingj. We find in the case of four particles the ratios 3.08(Y),
3.28(y),
3.302(y),
and in the case of six particles 2.47(-l?),
3.20(y).
The case of j = y is influenced strongly by the proximity of the half shell and does not conform too well to the predicted pattern [see ref. “)I. Even if only a small amount of pairing is present, the ratio of the excitation energies is quite different from the ideal rotational value. For instance for /I = 0.1 and j = 7 the ratio has dropped to a value of 2.32 for N = 4 and 2.48 for N = 6. From this point on it decreases more slowly towards the limit 1 in the pure pairing case.
M. VALLIBRES
252 QE
06
AND R. M. DREIZLER
:-/r;-_ ) =l7/2.N=6
04
0.2
-=;
O.9
I.0
)
0.0
Fig. 7. The quadrupole transition sum rule between the ground state and the J = 2 states for the configuration (g)6 as a function of the relative coupling strength. Note the scaling of the lower curve.
I
1= .C
j = l7/2.N=6
I.C
O.!
0.c
Fig. 8. The quadrupole transition sum rule between the first excited J = 2 state and the J = 4 states for the configuration (q)6.
COLLECTIVE
EFFECTS
283
3.2. TRANSITIONS
More detailed evidence about the “collectivity” of states can be gleaned from the transition rates. For the quadrupole transitions we offer the following remarks. The sum rule s’,Zd(G,x)
=
c B2(2%o>‘,
(31)
Y
for the E2 rates starting with the ground state is (nearly) exhausted by the transition to the lowest 2+ state, regardless of the relative coupling strength. In the x = 0 limit the lowest 2+ state has seniority two, all the other 2+ states having seniority four. The strength of the transition rate to the lowest 2+ state does not increase markedly with increasing quadrupole force component, that is at least for the range of j-values considered here. The situation is illustrated for the configuration (q)” as a representative example in fig. 7. Fig. 8 (using the same configuration as an example) shows the situation for the sum rule S!?(G, X) = c B,(4y, 2)‘, (32) Y
TABLE 2a
The stripping amplitudes for selected transitions between states of the configurations &U?‘(6),
Zr = l(4))’
2.910 0.000 0.000 0.000 0.000 0.000
2.531 0.148 0.110 0.014 0.009 0.003
1.048 0.623 0.066 0.270 0.707 0.017
A4(41,01)2 A4(42, 01)2 /t~(44,01)2 A+%, 01)2 A4(4y, 01)2
5.250 0.000 0.000 0.000 0.000
4.588 0.688 0.041 0.014 0.181
0.255 3.377 1.365 0.535 0.872
&(21,21)” A&23, 21)2 _4,,(24, 21)”
1.333 0.000 0.000 0.000
1.206 0.011 0.003 0.055
0.667 0.015 0.202 0.169
A&,
6
~26
-y =
3,
~16 -y=2.5
5,-l
Ao@Y,
rs -7 =
0112
0112
2112
&(Ol, 21y Az(02,21)2 3
A&J,
=
4
A2mJ,
* See table 2b.
0.417
0.388 0.59a* 0.003*
0.209 0.447 0.046
0.001 0.426 0.091
0.320
0.291 2.502* 0.054* 0.009*
0.076 2.536 0.096 0.076
0.081 1.549 0.307 0.369
21)2
.42(21,21y /&(22,21)’ A,(23,21)* ~16 Al
@=O 0.938 0.418 0.078
Ao@?J,
=
a=@=1 2.209 0.057 0.007
.4,(21,01)2 A,(22,01)2 A,(23,01)’ z&(24, 01)2 &(25,01)’ -716 dy
a =O.l
2.333 0.000 0.000
&(Ol, 01y &(02,01)2 ~8 dY=3
a=0
(J$)4 + (9)”
2112
M. VALLII?RES
284
AND R. M. DREIZLER
the E2 rates starting from the lowest 2f state to the 4’ states. In the seniority limit the transition rates to the degenerate 4’ states of seniority 4 are of comparable strength and are larger than the transition rate to the state with 1 = 4, s = 2. As soon as some mixing among these basis states is introduced by the Q-Q component, the TABLE 2b
The pick-up ampiitudes 4(Zl(6),
for selected transitions between states of the configurations
J’yi4))’
a=0
a = 0.1
cc=@=1
(%I6 + (-!J)4 B=O
Ao(Ol,ol)* Ao(Ol, 02)2 Ao(Ol,O3)2
2.333 0.000 0.000
2.209 0.008 0.000
0.938 0.088 0.003
A,(Ol, 21)2 &(Ol, 2212 /&(01,23)* A2(01,2Y>2
0.417 0.000 0.000 0.000
0.209 0.001 0.005 0.000
0.001 0.122 0.247 0.016
A4(01,41)2 &(Ol, 42)2 A4(01, 4y)Z
0.750 0.000 0.000
1.237 0.044 0.003
1.315 1.327 0.179
Ao(21,21)2 Ao(21, 23)2 Ao(21, 2Y)2
1.333 0.000 O.ooO
1.206 0.023 0.003
0.667 0.070 0.006
A1(21,01)2 &(21,02)2 A&21,03)*
2.917 0.000 0.000
2.513 0.102 0.000
1.408 0.372 0.012
A2(21,21)2 Az(21,22)2 A,(21, 23)2 &(21,2Y)s
0.320
0.290 0.116* 0.007* 0.02s
0.078 0.036 0.022 0.000
0.080 0.002 0.166 0.016
A1(21,41)2 x42(21,42)’ A&x 4Q2 A2ca 4rY
1.314
1.339 0.476* o-079* 0,063
1.796 0.100 0.002 0.015
0.784 0.695 0.485 0.055
* We give the values for a = 0.1, as there is degeneracy for a = 0.
main strength is concentrated in the lowest of the dominantly seniority-4 states. As the effect of the Q-Q correlations increases, the first excited 4+ state (dominantly seniority 2) picks up strength, and in the ‘?otational limit” this state as a member of the ground state band practically exhausts the sum rule. Tables 2a and b show the spectroscopic amplitudes for two-particle stripping and pick-up processes among states of the configurations (y)” and (y)“. The spectroscopic amplitudes are conventionally defined as
(33)
COLLECTIVE EFFECTS
285
for the stripping process and sk(l;,Yi(N
+
hYf(N-2)) = C (-Jk-” 4
[M:+~ “,:J x
(zrM,y,(N-2)IA:l~iM,+qyi(N)),
(34)
for the pick-up process. For comparison of the cross section to final states with the same angular momentum, it is only necessary to list the square of the reduced matrix elements (see definition in eqs. (17) and (18)). The first three sections in each table contain the amplitudes for the direct transfer that start with the ground state of the respective target and populate final states with spins 0, 2 and 4. In the limit x = 0 the amplitudes are governed by seniority selection rules. As one turns on the Q-Q interaction, one observes a distribution of both the stripping and pick-up strength to various final states. The pattern of strength distribution that emerges in the rotational limit G = 0 is similar for each ‘of the j-values studied. The tables also include some of the amplitudes involved in (core excitation) two-step processes 01(N) --, Z?(N) + I’y’(N-2), Ol(N-2)
a
EXACT
Q
VARlATlONAL
6
RPA
--, Iy(N-2)
+ I’y’(N).
Fig. 9. Comparison of the excitation energy of the first excited J = 2 state from the variational calculation with the exact result as a function of the relative coupling strength for the configuration (y)“. The format corresponds to fig. 1.
M. VALLIERES
286
AND
R. M. DREIZLER
It might be worthwhile to investigate the importance of two-step processes in terms of this model more closely. 3.3. COMPARISON
WITH
VARIATIONAL
CALCULATION
In comparing the excitation energies of the first excited 2+ state calculated by the simple variational approach with the exact results [see fig. 9 for the case (y)“] we find that the general trend is reproduced over the complete range of coupling constants. The detailed agreement is not perfect in the “transition” region (a M j? z I), but it is quite good near the two limiting situations (X = 0 and G = 0). We also find better agreement with the exact results for increasingj and a given particle number N. For comparison we have also indicated the results of the standard random-phase approximation, which fails beyond a definite value of x/SsZG. In fig. 10 we compare the four basic amplitudes of the variational approach A,(OO), &(20), A,(20) and A,(02) with the exact results. The cases plotted are for the configuration (y-)->”as an illustrative example. The assumptions made in the variational calculation are exact in the seniority limit. We note that the difference A,(20) A,(02) is calculated quite accurately by the variational principle, although the individual quantities differ appreciably from the exact results in the rotational limit. This explains the relatively good results for the excitation energy [see eq. (30)]. In order to explore the assumption of one state saturation of the sum rules (23)(25) we have plotted in fig. 11 the right- and left-hand sides of these equations for the
Fig.
10. Comparison
of the amplitudes involved in the variational result for the configuration (q)“.
calculation
6, with the exact
COLLECTIVE EFFECTS
287
Fig. Il. The right-hand side and left-hand side of the subsidiary conditions [eqs. (23)-(25). (27)] is evaluated with the assumption of one state saturation. The con6guration is (9)“. In the case of the Casimir operator [eq. (25)] we also give the left-hand side evaluated with all the intermediate states.
example of(y)“. For the right-hand side we give the exact value, for the left-hand side we use only one state in the intermediate sum with the amplitudes obtained from the exact calculation. For the Casimir operator [eq. (25)] we also give the left-hand side with all possible intermediate states. The difference in the curves labelled RHS and LHS(l) then indicates the error introduced by the additional assumption of preferential quadrupole correlations in the ground state. The assumption of one state saturation is less justified for the additional relations (26)-(29). As an example we present in fig. 11 the situation for eq. (27). However, one can obtain more accurate relations for the computation of the additional amplitudes by using combinations of the equations, as for instance the sum of eqs. (28) and (29). 4. Concinsion
The model is not sufficiently realistic to invite comparison with the real world. Among other things the “phase transition” to the deformed solution occurs for rather “unrealistic” coupling strengths. However it provides insight into the basic mechanisms. One observes the emergence of rotational (like) bands in the limit of strong quadrupole forces. This is accompanied by a respective adjustment of the multipole
288
M. VALLIERES
AND R. M. DREIZLER
transition and particle transfer strengths among the low-lying states. It is perhaps interesting to note that these matrix elements indicate a strong-coupling situation before the energies of the ground state band adjust fully to the (approximate) Z(Z+ 1) law. One finds a possible parallel in recent experimental information on quadrupole matrix elements in vibrational nuclei. A problem, that may be of interest, is the investigation of the competition of direct and two-step processes as a function of the relative coupling strengths in terms of this model. The variational calculations of ref. “) for this model are very simple It is gratifying that agreement with the exact solution is quite reasonable for the complete regime of coupling strengths. This agreement certainly warrants an extension of this approximation scheme to more realistic situations. The comparison with exact results also contains a word of caution. The formulation of approximations in the variational approach or the equation-of-motion formalism of refs. ‘, “) require insight into the situation near both limiting cases x = 0 and G = 0. The subsidiary sum rules used in the variational approach as well as the equations of motion should be treated with sufficient care, if intermediate sums are saturated with few states. For example, the patterns found for the sum rule (32) shows that two “different” 4+ states dominate this sum rule in the two limits and correspondingly two 4’ states are required in the intermediate coupling region (x/5QG z 1) for a reasonable approximation. In handling the subsidiary conditions (or equations of motion) it is worthwhile to consider combinations to give for example the quantity [A,(20)-A,(02)] rather than the individual amplitudes. We wish to thank Drs. A. Lande and B. Bayman for making their CFP code available throughDr. K. Lips of the University of Kentucky, Lexington, Ky. We thank Dr. A. Klein for discussions and helpful comments. References 1) M. Baranger and K. Kumar, Nucl. Phys. 62 (1964) 113 2) T. Marumori, Y. Shono, M. Yamamura and Y. Miyanishi, Phys. Lett. 25B (1967) 249; Report in the Int. Conf. on nuclear structure, XIII, V, Tokyo, 1967 3) T. Marumori, M. Yamamura, Y. Miyanishi and S. Nishiyama, Prog. Theor. Phys., Suppl. extra number (1968) p. 179 4) D. H. E. Gross and M. Yamamura, Nucl. Phys. A140 (1970) 625 5) Belyiav, Nuclear structure Dubna Conf. (IAEA, Vienna, 1968) 6) R. M. Dreizler and A. Klein, Phys. Lett. 3OB (1969) 236; Proc. of Midwest Conf. on theoretical physics, Notre Dame University, 1970, to be published 7) W. J. Mulhall and L. Sips, Nucl. Phys. 57 (1964) 565 8) M. I. Friedmann and I. Kelson, Nucl. Phys. Al44 (1970) 209 9) A. Lande and B. Bayman, Nucl. Phys. 77 (1966) 1 10) A. M. Lane, Nuclear theory (Benjamin, New York, 1964) 11) M. Harvey in Advances in Nuclear Physics, vol. 1, ed. M. Baranger and Vogt (Plenum Press, New York, 1968) 12) A. Arima, Prog. Theor. Phys., Suppl. extra number (1968) p. 489 13) A. Hassitt, Proc. Roy. Sot. 229 (1955) 110
@
~o?tb-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint oc microfilm without written permission from the publisher
COLLECTIVE EFFECTS IN THE j” CONFIGURATION M. VALLIERES
t and R. M. DREIZLER
*t
Department of Physics, University of Pennsylvania, Phi~ade~hia, Pa. 19104, USA Received 10 May 1971 (Revised 2 July 1971) Abstract:
Using a model Hamiltonian with pairing and quadrupole forces we investigatethe structure of selectedstates of the cotiguration j” withj = Idp,3, T and n = 4,6 as a function of the reIativecouplingstrength. Exact results for excitation energies, E2 transition rates and twoparticle transfer spectroscopic amplitudes are presented and compared with the results of pre-
vious approximate variational calculations for the same model.
1. Introduction Collective effects in nuclei have been attributed to the dominance of the long-range components of the residual nuclear forces over the short-range part. Several simplified models have been suggested for the investigation of the underlying many-body aspects. Among these, the model using a single&shell and a pairing plus quadrupolequadrupole Hamiltonian offers a convenient starting point for more ambitious microscopic descriptions. This model has been used in particular as a testing ground for new approaches to the many-body problem. These methods are characterized by the direct calculation of physically observable quantities such as multipole matrix elements and spectroscopic amplitudes for particle transfer without the explicit in~rvention of wave functions. A main tool is the exploitation of sum rules for these matrix elements. So far, these efforts have concentrated on the emergence of rotational characteristics in the pure quadrupole limit 2- “) an d on the behavior of the properties of low-lying states in the transition from the pure pairing to the pure quadrupole interaction “). The present investigation has been motivated largely by the desire to compare the results of the simple scheme of ref. “) with exact solutions. These are available for four- ‘9“) (j = $-J$) and six- *) (1 = 9, -?i_) particle systems, but the main emphasis in these investigations was centered on energy spectra. With the emphasis on collective features, we are also interested in the transition amplitudes to states of the same and neighboring nuclei. In addition “collective effects” are more pronounced for larger j-values and particle numbers (that do not approach the mid-shell point N = _)(2j+ 1)). t Supported by a stipend from the Ministbre de I’Education, Quebec, Canada. tf Supported in part by the US Atomic Energy Commission. 272
COLLECTIVE EFFECTS
273
We have thus attempted to provide exact results for as large a system j” as was accessible with our computer facilities. This turned out to be the configuration (y)“. The procedure used is the standard shell-model technique. We calculate singleparticle coefficients of fractional parentage in the seniority basis using a modified version of a code provided by Bayman and Lande ‘). We then set up the Hamiltonian matrix and diagonal& it. Relevant details are contained in sect. 2, together with a description of the model and a brief resume of the variational method of ref. “). In sect. 3 we present the exact results and compare them with those obtained by the variational method. In all we place emphasis on the variation of the results with the relative strength of the force components, thej-value (y, 9, y) and particle number (N = 4,6). 2. Theory 2.1. THE MODEL
The model Hamiltonian contains a pairing part with the force constant G and a quadrupole part with the strength x. We write it in the form H = -G~A~+A~-~xC(-)~B,~B~_~, Q Q = j+*.
(1)
Single-particle terms are not relevant in this problem. The operators A:+, A: and Bi are special cases of the general two-particle transfer and multipole operators that
can be constructed from the single-shell fermion creation and destruction operators
B: = (2k+1)-*C
m
[A qjm
k-J (-)i-n’+%m+am_q.
These operators can be shown to span the group 0(4j+2) with the Lie algebra [g,
A$+] = ~2J2k’+l(-)2’-k”
[; ;: ,“;;,I(::
5’ ;“) Akq:q,,
k”
[A;, A;:+] =
a,,,6,,
- 2 c ( -)2i+q[(2k+
1)(2k’+ 1)(2k”+ I)]+ X [
[B;, B::] = g ( -)2j-k”[1
(4)
_;
;:
,r”l’,] (; :’ :“)Bfz:‘_, ,
(5)
_( _)k+k’+k”]
xJ2k”+l
[;
;: ,f”;‘,1(;
:’ y)B:;,,.
(6)
We have used the notation [ ] for Clebsch-Gordan coefficients and { } for 6-j symbols.
274
M. VALLIBRES AND R. M. DREIZLER
In the pure pairing limit (x = 0) we only have to consider the SU(2) subalgebra spanned by the operators A:+, Ai and the number operator N = (2j+ 1)*I$. This leads to the well-known seniority scheme lo). In the pure quadrupole limit (G = 0) we have a situation reminiscent of the SU(3) scheme ‘I). If we consider the commutation relations of the operators B,’ and Bt, we find
(8)
Eq. (7) identifies Bi as the angular momentum operator B: = [j(j+l)(2j+l)]+Jq;
(10)
however the identifi~tion of Bi as the SU(3) quad~pole tensor is spoiled by the appearance of the Bi term in the commutator (9). Arima 12) has shown that the typical rotational band structure of the SU(3) scheme is preserved for states IJMa), provided that J < 2j-t 1 and j B 1. 2.2. THE EXACT
SOLUTION
For the exact solution of the problem we first construct a set of basis states in terms. of the coefl?cient of fractional parentage (c.f.p.) in the seniority limit
The antisymmetric iv-particle state Y(jNsc&W), classified by seniority s, anguiar momentum I, M and a counting label CL,is expressed in terms of the angular momentum coupled set of (N- I)-particle and one-particle states. In the language of second quantization, the c.f,p. are defined by the relation
The matrix elements of the Hamiltonian operator (1) in the basis &V)sollM> can be evaluatedin terms of the c.f.p. connecting the IV-and (N- l)-particle systems and the c.f.p. connecting the (N- I)- and (N-2)-particle systems. The latter are defined as <(N-l)vBJ~la,tI(N-2)salM)
= x/N- 1
(131
275
COLLECTIVE EFFECT‘S
The resulting expression for the pairing and quadrupole interactions is ((N)saZMIA~+A~((N)s’a’ZM)
= ~3,~N(N-l)
4Q(2(21+1)
-N(N-1)
x Jvfl ,
J’v’p’. I”a”S”
x u~~~~~~~~~(Jv~)u I’,~,,ct”(J’v’s’)vJ,y,~,~Zs’a’).
(15)
Here the curly bracket stands for a 9-j symbol. The calculation of the c.f,p. in the seniority limit was performed with a code provided by Bayman and Lande ‘). The code uses the fact that the basis states I(N)s belong to the representation [l, F,r ‘,, I] of SU(2j+ 1) and transform irreducibly under Sp(2j+ 1). If one evaluates the Casimir operators (G) of both groups in the basis of non-antisymmetric product states PW
- ~)V~~~)~(~)l~,
and diagonaiizes the resulting matrices, the physical eigenstates can be identified by inspection of the eigenvalues. The c.f.p. are then given by the appropriate eigenvectors of the matrix that diagonalizes an arbitrary combination a(G(SU(2j+
1))) +b(G(Sp(2j+
1))).
A convenient check of the calculated c.f.p. is provided by eq. (14). The pairing part of the Hamiltonian is diagonal with respect to the basis (11) (in the counting label a enforced by orthogonalization). The main limitation of this method lies in the dimensionality of the (over complete) set of states [Y(N-- 1)4]5. In table la we give the dimensionality of the product basis TABLE la
The number of basis states for the case of six particles as a function of angular momentum J and
the shell label j
I=0
I=2 x=4
9
9
21
23
T
YE-
zs z
19( 8) 89(16) 153(26)
26(10) 147(23) 194(37)
36(13) 173(31) 302(S)
47(16) 230(41) 317(68)
61(20) 296(53) 409(88)
The number of physical states with symmetry (1, r. N. + . ., I] is given in brackets.
27
7
77( 24) 378( 67) 523(112)
276
M. VALLIBRES AND R. M. DREIZLER
together with the number of physical states with angular momentum 0,2,4 of selected six-particle problems. An alternative method for the calculation of the c.f.p. has been suggested by Hassitt 13). With the c.f.p. we can calculate the Hamiltonian matrix using eqs. (14) and (1.5). After diagonalization the eigenvectors are of the form
The individual matrix elements of the multipole and transfer operators Z$ and Ai f for the eigenstates (16) are obtained by the following- steps: We first calculate the reduced matrix elements in the seniority basis, using the definition ((N)ZMs,IB~I(N)Z’M’,‘cr’) = [2Z + l]-* ~(~)Z~~~l~~~l~~-
Ml
“, k ’
CM 4
B,((N)Zsa, (N)Z’s’a’),
2)Z’M’s’a’) = [(l --Sk, e)2Z$ I]-+ A,((N)Zsa, (N - 2)Z’s’a’).
The results are B,((N)ZSol, (N)Z’s’cL’)= N[(2z+ 1)(2i’+ 1)]3( --y-r
~~(~~)ZS~,(~-~~~'S'~') = (-)k""-"[&N(N- 1)(2k+1)(21+1)]
Next, the reduced matrix elements among the eigenstates are obtained by Z3&, I’?‘) = C C~y~~(Zsa, I’s’a’)C$f?, se,S’.’
A&y, I’y’) = 2.3. VARIATIONAL
c CE’yAk(Zsa, Z’s’a’)C~~72)r’y’. SC& S’OI’
METHOD
In this section we give a brief summary of the variational calculation proposed by Dreizler and Klein “). The ground state expectation value of the IIamiltonian is evaluated under the assumption that the sums over intermediate states occurring in the matrix elements (OOlA~+ -4~100) and (OOl’&( --r Bt B!, lO0) are saturated by only one state
<~ol~oo+4l~> = r: l&(O, WI2 = l4(O@),
qN4))l:=,
= c lB,(2y, 0)l” a IZ3,(2,0)1&~ * I
,
(21) 122)
COLLECTIVE EFFECTS
271
The gro~d-state-to-ground-state two-particfe transfer amplitude A ~~~~ and the quadrupofe transition matrix element to the first excited state of angular momentum two &(2,0) are calculated by a variational principle with the following kinematical restrictions (A,&, 1’) = A,(I(N--21, l’(N-4)):
The first two equations are obtained from the afgebra, eqs. (4) and (5) with k = fc’ = 2, under the same ~surn~io~s as for the eateulation of the ground state energy. The last equation is derived from the Casimir operator of the full group 0(4i+ 2) under the czdditroml assumption that correlations of multipole order greater than two can be neglected for its ground state expectation value, With the four amplitudes Ae(0, 0), A2(2, 0), A2(0, 2) and ,S,(2,0) obtained from the ground state variational principle, we can calculate additional ones f&(2,2), J&(2,2>, M2,2)3 using the sum rules (241 4(2, @4,(2* 2)-Q%<@ 2)4.(2,2} 4@, 2&<(2,
A,@,
O)B&
= 50 (5 ‘i ;) I&(2,
O),
(27)
q--B,f2,O)f%z(2,2) = 10 (5 ; ;) A&O, 21,
2j--B,<@
0~~2~22~
=
-
10
;
;
;
t
A2@,
0).
cw
I
These sum rules are also consequences of the 0(4j+2) algebraand the assumption of one state saturation in the intermediate state summation. In addition, one can obtain an approximate expression for the excitation energy OFthe first excited state with angular moments two
%o = from the commutator
E,--E,,
[.B&Hj. This yields
9 X+@ A*(O*0) B(2 a20 = Sj(j + 1)(2j + 1) 2 3
fA2@9
0)
-A&h
211.
P9
As stated in the introductions one of our objectives is the comparison of the results
278
M. VALLIeRE.3
AND R. M. DREIZLER
x
00
0
a=56a 2 4
(x/51
Fig. 1. The spectrum of the J = 0,2, 4 states of the configuration (+)’ as a function of the relative coupling strengths c( = x/SQG -+ 0 to 1 and p = 5GQ/x + 1 to 0. The energies are scaled to
-ix
x/5
Fig. 2. The same as fig. 1 for the configuration (J#)6. For clarity we have not included all the highly excited states.
279
COLLECTIVE EFFECTS
-ai
s=o
(‘7/d
-3.0 -
I
Fig. 3. The same as fig. 2 for the configuration (J$)4.
Fig. 4. The same as fig. 2 for the configuration (AJ)‘j.
280
M. VALL@RES
AND R. M. DREIZLER
2 4
-3.0 -
Fig. 5. The same as fig. 2 for the configuration
(9)‘.
x/5
I
Fig. 6. The same as fig. 2 for the configuration
(y)“.
COLLECTIVE
281
EFFECTS
of this variational calculation with the exact solutions. In addition, we would like to check the quality of the saturation of the various sum rules with one intermediate state. This, together with a discussion of the exact results in their own rights, will be presented in sect. 3. 3. Results 3.1. ENERGY
SPECTRA
We have performed the calculations outlined in sect. 2.2 for states with I = 0, 2, 4 in the configurations (y)4, (y)6, (J$)4, (y)” and (q)4, as well as for states with I = 0, 2 in the configuration (y)“. The spectra are shown in figs. l-6 as a function of the relative strength of the two parts of the interaction. The relative strength is TABLE lb The number of physical states of given angular momentum in each configuration
State
8 N=4
I=0
I=2 I==4
9
9 N=6
N=4
N=6
N=4
N=6
4 7 11
3 5 7
8 16 26
4 6 9
13 31 51
2 4 5
-
measured by a = x/5GQ on the pairing side and by j?l = 5GSZ/x on the quadrupole side. Note also that the energies are given in units of QG on the pairing side and in units of 3x on the quadrupole side. For clarity we have not included in these figures all the highly excited states. The total number of states for each angular momentum are listed in table lb. We ,remark the following: in the limit of a pure quadrupole force (/I = 0) we expect the lowest state of each angular momentum to form a rotational band. According to Arima “),the ratio of excitation energies R4 = (E4-Eo)/ (E, -E,) should approach the ideal value of q with increasingj. We find in the case of four particles the ratios 3.08(Y),
3.28(y),
3.302(y),
and in the case of six particles 2.47(-l?),
3.20(y).
The case of j = y is influenced strongly by the proximity of the half shell and does not conform too well to the predicted pattern [see ref. “)I. Even if only a small amount of pairing is present, the ratio of the excitation energies is quite different from the ideal rotational value. For instance for /I = 0.1 and j = 7 the ratio has dropped to a value of 2.32 for N = 4 and 2.48 for N = 6. From this point on it decreases more slowly towards the limit 1 in the pure pairing case.
M. VALLIBRES
252 QE
06
AND R. M. DREIZLER
:-/r;-_ ) =l7/2.N=6
04
0.2
-=;
O.9
I.0
)
0.0
Fig. 7. The quadrupole transition sum rule between the ground state and the J = 2 states for the configuration (g)6 as a function of the relative coupling strength. Note the scaling of the lower curve.
I
1= .C
j = l7/2.N=6
I.C
O.!
0.c
Fig. 8. The quadrupole transition sum rule between the first excited J = 2 state and the J = 4 states for the configuration (q)6.
COLLECTIVE
EFFECTS
283
3.2. TRANSITIONS
More detailed evidence about the “collectivity” of states can be gleaned from the transition rates. For the quadrupole transitions we offer the following remarks. The sum rule s’,Zd(G,x)
=
c B2(2%o>‘,
(31)
Y
for the E2 rates starting with the ground state is (nearly) exhausted by the transition to the lowest 2+ state, regardless of the relative coupling strength. In the x = 0 limit the lowest 2+ state has seniority two, all the other 2+ states having seniority four. The strength of the transition rate to the lowest 2+ state does not increase markedly with increasing quadrupole force component, that is at least for the range of j-values considered here. The situation is illustrated for the configuration (q)” as a representative example in fig. 7. Fig. 8 (using the same configuration as an example) shows the situation for the sum rule S!?(G, X) = c B,(4y, 2)‘, (32) Y
TABLE 2a
The stripping amplitudes for selected transitions between states of the configurations &U?‘(6),
Zr = l(4))’
2.910 0.000 0.000 0.000 0.000 0.000
2.531 0.148 0.110 0.014 0.009 0.003
1.048 0.623 0.066 0.270 0.707 0.017
A4(41,01)2 A4(42, 01)2 /t~(44,01)2 A+%, 01)2 A4(4y, 01)2
5.250 0.000 0.000 0.000 0.000
4.588 0.688 0.041 0.014 0.181
0.255 3.377 1.365 0.535 0.872
&(21,21)” A&23, 21)2 _4,,(24, 21)”
1.333 0.000 0.000 0.000
1.206 0.011 0.003 0.055
0.667 0.015 0.202 0.169
A&,
6
~26
-y =
3,
~16 -y=2.5
5,-l
Ao@Y,
rs -7 =
0112
0112
2112
&(Ol, 21y Az(02,21)2 3
A&J,
=
4
A2mJ,
* See table 2b.
0.417
0.388 0.59a* 0.003*
0.209 0.447 0.046
0.001 0.426 0.091
0.320
0.291 2.502* 0.054* 0.009*
0.076 2.536 0.096 0.076
0.081 1.549 0.307 0.369
21)2
.42(21,21y /&(22,21)’ A,(23,21)* ~16 Al
@=O 0.938 0.418 0.078
Ao@?J,
=
a=@=1 2.209 0.057 0.007
.4,(21,01)2 A,(22,01)2 A,(23,01)’ z&(24, 01)2 &(25,01)’ -716 dy
a =O.l
2.333 0.000 0.000
&(Ol, 01y &(02,01)2 ~8 dY=3
a=0
(J$)4 + (9)”
2112
M. VALLII?RES
284
AND R. M. DREIZLER
the E2 rates starting from the lowest 2f state to the 4’ states. In the seniority limit the transition rates to the degenerate 4’ states of seniority 4 are of comparable strength and are larger than the transition rate to the state with 1 = 4, s = 2. As soon as some mixing among these basis states is introduced by the Q-Q component, the TABLE 2b
The pick-up ampiitudes 4(Zl(6),
for selected transitions between states of the configurations
J’yi4))’
a=0
a = 0.1
cc=@=1
(%I6 + (-!J)4 B=O
Ao(Ol,ol)* Ao(Ol, 02)2 Ao(Ol,O3)2
2.333 0.000 0.000
2.209 0.008 0.000
0.938 0.088 0.003
A,(Ol, 21)2 &(Ol, 2212 /&(01,23)* A2(01,2Y>2
0.417 0.000 0.000 0.000
0.209 0.001 0.005 0.000
0.001 0.122 0.247 0.016
A4(01,41)2 &(Ol, 42)2 A4(01, 4y)Z
0.750 0.000 0.000
1.237 0.044 0.003
1.315 1.327 0.179
Ao(21,21)2 Ao(21, 23)2 Ao(21, 2Y)2
1.333 0.000 O.ooO
1.206 0.023 0.003
0.667 0.070 0.006
A1(21,01)2 &(21,02)2 A&21,03)*
2.917 0.000 0.000
2.513 0.102 0.000
1.408 0.372 0.012
A2(21,21)2 Az(21,22)2 A,(21, 23)2 &(21,2Y)s
0.320
0.290 0.116* 0.007* 0.02s
0.078 0.036 0.022 0.000
0.080 0.002 0.166 0.016
A1(21,41)2 x42(21,42)’ A&x 4Q2 A2ca 4rY
1.314
1.339 0.476* o-079* 0,063
1.796 0.100 0.002 0.015
0.784 0.695 0.485 0.055
* We give the values for a = 0.1, as there is degeneracy for a = 0.
main strength is concentrated in the lowest of the dominantly seniority-4 states. As the effect of the Q-Q correlations increases, the first excited 4+ state (dominantly seniority 2) picks up strength, and in the ‘?otational limit” this state as a member of the ground state band practically exhausts the sum rule. Tables 2a and b show the spectroscopic amplitudes for two-particle stripping and pick-up processes among states of the configurations (y)” and (y)“. The spectroscopic amplitudes are conventionally defined as
(33)
COLLECTIVE EFFECTS
285
for the stripping process and sk(l;,Yi(N
+
hYf(N-2)) = C (-Jk-” 4
[M:+~ “,:J x
(zrM,y,(N-2)IA:l~iM,+qyi(N)),
(34)
for the pick-up process. For comparison of the cross section to final states with the same angular momentum, it is only necessary to list the square of the reduced matrix elements (see definition in eqs. (17) and (18)). The first three sections in each table contain the amplitudes for the direct transfer that start with the ground state of the respective target and populate final states with spins 0, 2 and 4. In the limit x = 0 the amplitudes are governed by seniority selection rules. As one turns on the Q-Q interaction, one observes a distribution of both the stripping and pick-up strength to various final states. The pattern of strength distribution that emerges in the rotational limit G = 0 is similar for each ‘of the j-values studied. The tables also include some of the amplitudes involved in (core excitation) two-step processes 01(N) --, Z?(N) + I’y’(N-2), Ol(N-2)
a
EXACT
Q
VARlATlONAL
6
RPA
--, Iy(N-2)
+ I’y’(N).
Fig. 9. Comparison of the excitation energy of the first excited J = 2 state from the variational calculation with the exact result as a function of the relative coupling strength for the configuration (y)“. The format corresponds to fig. 1.
M. VALLIERES
286
AND
R. M. DREIZLER
It might be worthwhile to investigate the importance of two-step processes in terms of this model more closely. 3.3. COMPARISON
WITH
VARIATIONAL
CALCULATION
In comparing the excitation energies of the first excited 2+ state calculated by the simple variational approach with the exact results [see fig. 9 for the case (y)“] we find that the general trend is reproduced over the complete range of coupling constants. The detailed agreement is not perfect in the “transition” region (a M j? z I), but it is quite good near the two limiting situations (X = 0 and G = 0). We also find better agreement with the exact results for increasingj and a given particle number N. For comparison we have also indicated the results of the standard random-phase approximation, which fails beyond a definite value of x/SsZG. In fig. 10 we compare the four basic amplitudes of the variational approach A,(OO), &(20), A,(20) and A,(02) with the exact results. The cases plotted are for the configuration (y-)->”as an illustrative example. The assumptions made in the variational calculation are exact in the seniority limit. We note that the difference A,(20) A,(02) is calculated quite accurately by the variational principle, although the individual quantities differ appreciably from the exact results in the rotational limit. This explains the relatively good results for the excitation energy [see eq. (30)]. In order to explore the assumption of one state saturation of the sum rules (23)(25) we have plotted in fig. 11 the right- and left-hand sides of these equations for the
Fig.
10. Comparison
of the amplitudes involved in the variational result for the configuration (q)“.
calculation
6, with the exact
COLLECTIVE EFFECTS
287
Fig. Il. The right-hand side and left-hand side of the subsidiary conditions [eqs. (23)-(25). (27)] is evaluated with the assumption of one state saturation. The con6guration is (9)“. In the case of the Casimir operator [eq. (25)] we also give the left-hand side evaluated with all the intermediate states.
example of(y)“. For the right-hand side we give the exact value, for the left-hand side we use only one state in the intermediate sum with the amplitudes obtained from the exact calculation. For the Casimir operator [eq. (25)] we also give the left-hand side with all possible intermediate states. The difference in the curves labelled RHS and LHS(l) then indicates the error introduced by the additional assumption of preferential quadrupole correlations in the ground state. The assumption of one state saturation is less justified for the additional relations (26)-(29). As an example we present in fig. 11 the situation for eq. (27). However, one can obtain more accurate relations for the computation of the additional amplitudes by using combinations of the equations, as for instance the sum of eqs. (28) and (29). 4. Concinsion
The model is not sufficiently realistic to invite comparison with the real world. Among other things the “phase transition” to the deformed solution occurs for rather “unrealistic” coupling strengths. However it provides insight into the basic mechanisms. One observes the emergence of rotational (like) bands in the limit of strong quadrupole forces. This is accompanied by a respective adjustment of the multipole
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AND R. M. DREIZLER
transition and particle transfer strengths among the low-lying states. It is perhaps interesting to note that these matrix elements indicate a strong-coupling situation before the energies of the ground state band adjust fully to the (approximate) Z(Z+ 1) law. One finds a possible parallel in recent experimental information on quadrupole matrix elements in vibrational nuclei. A problem, that may be of interest, is the investigation of the competition of direct and two-step processes as a function of the relative coupling strengths in terms of this model. The variational calculations of ref. “) for this model are very simple It is gratifying that agreement with the exact solution is quite reasonable for the complete regime of coupling strengths. This agreement certainly warrants an extension of this approximation scheme to more realistic situations. The comparison with exact results also contains a word of caution. The formulation of approximations in the variational approach or the equation-of-motion formalism of refs. ‘, “) require insight into the situation near both limiting cases x = 0 and G = 0. The subsidiary sum rules used in the variational approach as well as the equations of motion should be treated with sufficient care, if intermediate sums are saturated with few states. For example, the patterns found for the sum rule (32) shows that two “different” 4+ states dominate this sum rule in the two limits and correspondingly two 4’ states are required in the intermediate coupling region (x/5QG z 1) for a reasonable approximation. In handling the subsidiary conditions (or equations of motion) it is worthwhile to consider combinations to give for example the quantity [A,(20)-A,(02)] rather than the individual amplitudes. We wish to thank Drs. A. Lande and B. Bayman for making their CFP code available throughDr. K. Lips of the University of Kentucky, Lexington, Ky. We thank Dr. A. Klein for discussions and helpful comments. References 1) M. Baranger and K. Kumar, Nucl. Phys. 62 (1964) 113 2) T. Marumori, Y. Shono, M. Yamamura and Y. Miyanishi, Phys. Lett. 25B (1967) 249; Report in the Int. Conf. on nuclear structure, XIII, V, Tokyo, 1967 3) T. Marumori, M. Yamamura, Y. Miyanishi and S. Nishiyama, Prog. Theor. Phys., Suppl. extra number (1968) p. 179 4) D. H. E. Gross and M. Yamamura, Nucl. Phys. A140 (1970) 625 5) Belyiav, Nuclear structure Dubna Conf. (IAEA, Vienna, 1968) 6) R. M. Dreizler and A. Klein, Phys. Lett. 3OB (1969) 236; Proc. of Midwest Conf. on theoretical physics, Notre Dame University, 1970, to be published 7) W. J. Mulhall and L. Sips, Nucl. Phys. 57 (1964) 565 8) M. I. Friedmann and I. Kelson, Nucl. Phys. Al44 (1970) 209 9) A. Lande and B. Bayman, Nucl. Phys. 77 (1966) 1 10) A. M. Lane, Nuclear theory (Benjamin, New York, 1964) 11) M. Harvey in Advances in Nuclear Physics, vol. 1, ed. M. Baranger and Vogt (Plenum Press, New York, 1968) 12) A. Arima, Prog. Theor. Phys., Suppl. extra number (1968) p. 489 13) A. Hassitt, Proc. Roy. Sot. 229 (1955) 110