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On schematic interactions in solvable models, especially the quadrupole pairing interaction
1.C
[
Nuclear Physics A243 (1975) 125--142; @) North-ltollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON SCHEMATIC INTERACTIONS IN SOLVABLE MODELS, ESPECIALLY THE QUADRUPOLE PAIRING INTERACTION J. K R U M L I N D E Department of Mathematical Physics, Lund Institute of Technology, S-220 07 Lund 7, Sweden Received 28 November 1974 Abstract: TJ~e properties of and the interplay between the monopole pairing, cluadrupole pairing
and quadrupole particle-hole interactions are studied in two simple, solvable models. i. Introduction The study of schematic interactions and the exploration of their properties by means of exact solutions in solvable models plays an important role in the understanding of nucleonic correlation effects in medium and heavy nuclei. Especially simple and useful interactions are, for instance, the ordinary (monopole) pairing interaction and the ordinary (single-particle or particle-hole type) quadrupolequadrupole interaction. These have been studied extensively previously l - s ) and their properties are well established throughout the whole range of values of the coupfing constants. The quadrupole pairing interaction is an interesting hybrid of pairing and quadrupole as it introduces correlations of quadrupole type in the particle-particle (pairing) component of the nuclear two-body interaction. As has been demonstrated recently [refs. 6-8)] it can offer an explanation of effects which were difficult to describe with only ordinary pairing and quadrupole interactions, such as the state-dependent pairing gap, introducing asymmetry between (t, p) and (p, t) cross sections 6, 7), and attenuation of Coriofis matrix elements in rotational states s, 9). This paper will try to describe the properties of the quadrupole pairing interaction and to investigate the effects of the interplay of the three schematic interactions. We also show how these interactions emerge in a natural way and thus yield the basic properties of nuclear interactions in some very simple solvable models. The influence of the quadrupole pairing interaction on rotational properties wilL however, not be discussed here as it is the subject of a separate paper 10). 2. Interactions in a multi-j-shell model Consider an n-level model, each level having, except for the twofold degeneracy due to time,reversal invariance of the Hamiltonian, an additional pair degeneracy ~2. 125
126
J. K R U M L I N D E
We introduce creation and annihilation operators for the levels ~ -- 1 . . . 2n, v = 1 . . . f2: eva, t cva. Further, we define single-particle operators, pair creation operators and pair annihilation operators: t c V=I
o
"t c~a,
(1)
V=l
see also refs. 11,12). The levels are labelled so that ~ = 2 n - l(m~ > 0) and ~ = 2n(m~, < 0) are time+ = J ' c ~+, 2 , - i ~ - --I . The operators defined in eq. (1) are the reversed partners, c~,2, generators of the rotation group R(4n). The transformation from the operators (1) to the R(4n) angular momenta L,p is given, for instance, in ref. 12), eqs. (44) and (45). If we assume that our model consists of f2 identical j-shells, the physical angular momentum is given by, assuming a basis withj~ diagonal: j_ = ~ x / ( i - 1 ) ( 2 n - i + 1)(N2~_ i , 2 i - 3 - N2i-2,2i)+ nN2n,2,-1, i=2 /1
j+ =j*--,
= i=l
n = j+½.
(2)
For the classification of the states of the n-level model the operators of the total pairing quasispin are of great systematic importance: L+ =
t=l
B'2,-1,2,,
L , = L*+,
(3)
/I
Lo = ½ Z (N2,-i,2,-I + N21,2i)-½nl2" f=l
Pairing quasispin and physical angular momentum form two commuting SU(2) algebras, together forming an R(4) algebra RLi(4). Thus R(4n) = RLj(4), and the operators of R(4n) can be classified as tensors with respect to the group
SCHEMATIC INTERACTIONS
127
RLj(4). We definetensor operators T~ in the followingway: [L+, T~] = (2(2 + 1 ) - #(/l +_1))'}T~ a,q,
ELo, Tu~] U_+,
=
#T~,
= (k(k + 1 ) - q(q_+
(4)
= qr¢ . The tensor operators fulfill the following conditions of hermiticity and time reversal: ( "~,~+qT T ~ t = ,.--., - - 2k" ~,-q,
~ - T ~ ---1 = (--)~+aT~a.
(5)
The commutation relations are of the form:
[T¢:::, r¢:2] = C(2, 2~ k~ kz) Z (2~ 2,- ~ ~ +~212~ ~ 22 ~2> Ak
xT~+~,q~+q~,
(6)
where C(2x22k~k2) is connected with the reduced matrix elements of the tensors T ~k. Eq. (6) is useful as it provides us with certain selection rules. It is easily seen that the operators defined by eq. (1) can be classified as RLj(4) tensors in the following way:
= {TlO +T°l-}-T~2+ TO3+ . . . + T 1,2i- l + TO,2i}. (v) Here T x° is the pairing quasispin: {N~,B~,B~p}~,~=l...2,,=21+l
7"+1°o = T-x/~-L_+,
Toi° = Lo,
(8)
TOo~ = j~.
(9)
and T °i is the physical angular momentum O1 T~+,
T~/~j+,
Tensors T ~k with 2 -- 1 will be called quasivectors, those with 2 = 0 quasiscalars. The quantum number k(or rather 2 ) is called the multipole order. A unitary subgroup Uo(2n) is formed by all # = 0 components of the T ~ tensors, i.e. all single-particle operators N~. Another subgroup of R(4n) - and also a subgroup of U0(2n) - can be formed out of all quasiscalar tensor components T o, z,-,. This is the symplectic group as employed by Flowers l~, ~ ) and others l s , ~ ) . It is here referred to as Sp~(2n). The operators belonging to this subalgebra commute with the total pairing quasispin operators, which is easily seen, e.g., from eq. (4). Thus, the pairing quasispin operators are symplectic invariants (invariant by transformations generated by Spr(2n ) generators) and, likewise, all Sp~(2n) operators are invariant by Bogolyubov-Valatin transformations, el. e.g. ref. 1s). The most general nuclear two-body interaction of the multi-j-shell model is composed of bilinear expansions in T~, forming rotationally invariant, hermitian combi-
128
J. K R U M L I N D E
nations of the the type type nations of '=°
T;,~T2~_~
= E
(_Y)"
v -"q -gq " = ~/2k+l ~
(10)
Taking 2 = 1, k = 0 gives the (ordinary) monopole pairing interaction, 2 = 1, = 0, k = 2 gives the quadrupole particle-hole interaction and 2 = 1,/~ = 1, lc = 2 the quadrupole pairing interaction. Actually, it is easily verified that t
j
Zff =
E
m,m'=-j
t
(ajmajm,,
¢t = I
{a~majm,, ~ .ajmajrn, ,
I.Z = 0 lot = - - 1 ,
using creation and annihilation operators labelled with the (j, m) quantum numbers. The gauge invariant pairing interaction 17. ~s) or surface delta interaction is given by 6 = E (2" 2k+ 1)-l[T)-=~kT#-:_~,]Jo=°. (12) k This is in general not a symplectic invariant 1s), since only T 1o commutes with all generators of Spe(2n). An exception from this rule is the multi-two-level model (j = ~) discussed below. The general form of the two-body interaction in this model is thus
V = 2 Z1,2 To°kaTo°~*+ 2 kodd
q
~.
Zk., 2 T1aakT~*:
keven ~t=--l, O, 1
(13)
q
If, further, Zk~is independent of/~ there is an additional degeneracy because V then commutes with L2, the total pairing quasispin. This condition is not fulfilled in the realistic cases, but such solutions provide us with interesting limiting cases which can be useful in the classification of states. 3. The multi-two-level model 3.1. I N T E R A C T I O N S
As is discussed, e.g. in Littlewood 19), the group R(8) possesses a group of six outer automorphisms, isomorphic with the symmetric group $3, interchanging irreducible R(8) representations of the same dimensions, cf. also Flowers and Szpikowski 2o). Thus it is possible to perform a non-trivial, unitary and canonical transformation of the generators L~a onto new generators J ~ , which are just as good as the old ones and more convenient for oar purpose. Note that this is possible only for R(8). See also ref. 12), subsect. 4.3, w ~ e such a transformation is given. In this new scheme the number-of-particles 0~6rator equals one of the generators, which turns out to be a great advantage. For convenience, table 1 gives the relations between the tensor operators, defined in sect. 2, and the g(8) generators, both in terms of the operators N~p, B~p, B~p and in terms of the specific R(8) operators r~. The direct relation between N~p, etc., and J~p is given, e.g. in ref. 11), table 1.
SCHEMATIC INTERACTIONS
129
TABLE 1 Definition of tensor operators
= ½ 4 3 ( J ~ 6 + s~ ~ ) - i K / 3 ( J ~ s + J67)
=
= -4~(s,+u,o) = -s~ is,7 + + 4 ~ ( s ~ - s ~ . )
=
x/3Ni2 ~/-~(i~+N**)
-
= N4z -- N~ 3 + ~/3Nz*
-~½43(A~-A~)
To°2
= ½(N , ~- N z ~ ) - + ( N ~ -
= 2Js7 +J6s
O3
N,,)
= J,5 - iJ47 - ½~/3(J56 - J 7 8 ) - i ½ 4 3 ( J 5 8 - J 6 7 ) = N31 - N v , -
To°32
= -4~(J48
= 4 ~ ( 1 % + N,~)
-/J,6)
7-o0_33
=
T~t~l
---- B~3
= - ½ ( J ~ - J 2 7 ) - i½(J25 + J xT)
---- B ~ , :T
=
TL~2
--45N2t
= ½(J26 - J 1 8 ) - - i½(S, 6 + Su8)
=
-4~(J2,-iJ14) --
~/3~43
t
= -4~W~-BL)
½(St 5 + J z T ) - i½(J2~
- -
= B~3
S17)
= +(s,~ + s ~ ) - ~ ½ ( s ~ - s ~ )
---- B~4
= - 4 ~T( J ~ 6 - * J ~ . )
= -~/T(sa7-iSa~ )
= - 4 ~ ( N , ~ + N~_0
J34
:/-oi_2, T~22
= ,
= ½(N11 + N22 - Na 3 - N44)
x[T(s37+iS~5) -~/~(J~6+U~.)
= 4~(N~, + N~) = 4 ~T( ,N, - N ~ 3 )
- ½ ( s ~ + s ~~ ) - ~½(s ~~ - J~.) 12 T-*it
= --824
= -- ½(Ja5 -q--J 2 7 ) + i½(J2s -- Sl 7)
= B23
12
= 4~(8I~-B3,)
7"1_~_ ~ = - ½ ( j ~ - z . ) + i½(A~ + z ~7)
= B14
12
T:~ _ ~ = - ~(S: ~ - J~ ~ ) - ~½(Z~~ + A ~) ToOl
= --813
i= - 4 ~T( J 5 6 - J7 s) - x/~(S67 - J58)
- 4~(J~ + iJ~7) -- -4~(s~-
N,~)-4~S~
ro°o~ T
T:oo To%o
=
= J12
10
T_~lo
4Ta_(j13+iJ23)
=
-4-~(J~3-iJ23)
T
+ 4 ~ ( J , ~ - ~J,~) = 4 ~ ( N ~ - N~,) + 4 ~ N , ~ _ -4~(B~2+B34) T ~ "t _
= ½(Nl~+N22+N33+N44)-f2
= 4~(B~2+B34)
130
J. K R U M L I N D E
The monopole pairing interaction takes the form Vz, = --Go ~rlO-r!Ot ~,o~,o = - - G o ( J 123 + J 22a ) ,
(14)
with Go -- Zo, the quadrupole pairing interaction VpQ =
--+GzZ
8
Ti12T:• "t =
-~G2 Z
2 -1-J2/I), 2 (J1/3
(15)
#=4
q
1 with -~G2 = )~z.1, and the quadrupole particle-hole interaction 8
VQ = ---~;g Z Tol~Tol~¢r = --~Z E S~#, q
(16)
.8=4
with @Z = Z2,o. The Flowers symplectic group Sp~(4) is generated by the three operators of T °I (physical angular momentum) and the seven operators of T °a (the quasiscalar octupole tensor). Its Lie algebra is isomorphic to the Lie algebra of a certain R(5) group, namely SpF(4) -~ RF(5) = {J~a; a, fl = 4, 5, 6, 7, 8}, (17) It is easily verified that the interactions Vp, Vt,Q and VQ, defined in eqs. (14), (15) and (16) respectively, commute with all the generators of RF(5). Thus all these interactions are sympleetic invariants, in contrast with what is the case for the general mu/ti-n-level model w i t h j ¢ ~. 3.2. T H E G E L ' F A N D BASIS A N D T H E P H Y S I C A L BASIS
The low-lying states span the spaces that carry the symmetric representations, el. ref. 1~), subsect. 3.10. As a basis we use the one defined by Gel'land and Ceflin 22). For the group R(8), such a basis state is, in Gel'land notation im 8 m7 m6 m5 m4 ma km2
0 0 0~ 0 0 0 0 0 , 0 0 0 0
(18)
here just shortened to
(m8 m7 m6 ms m,~ m3 m2).
(18')
The quantum number ms characterizes the representation itself. It is identical to ~2, the pair degeneracy. The dimension of the representation (~2) is given by de = ~6&~(I2+ 5)(~2 + 4)(0 + 3)2(0 + 2)(0 + 1). Now, the R(5) group in the chain R(8) = R(7) = . . .
= R(2),
(19)
SCHEMATIC INTERACTIONS
131
defining the Gel'fand scheme, is taken to be the Flowers symplectic group Spy(4) or RF(5). The quantum number F = ms,
(20)
characterizes the RF(5) representations (F0), the dimension of which is given by dr = I ( F + )(2F+3)(F+2).
(21)
Since RF(5)~Rj(3) where Ri(3) is the physical angular momentum group, we may change the basis from the one characterized by (m4m3m2)to a basis wherej z andj~ are diagonal, thus exchanging (m4mam2)for (JMsa) where ~ is an additional quantum number sometimes needed (see below). Further, due to R(8) ~ RL(3) X R~(5), we can change the basis so as to exchange the quantum numbers (LN), eigenvalues of the pairing quasispin operators L2 and L0.
(22)
(m7m6)for (LML) or
L21LML> = L(L + 1)[LMz>, LolLML) = MzILMr.),
(23)
N = 2Mz+212, where N denotes the number of particles. This is possible since L2 and Lo commute with the R(8) Casimir operators (and thus do not change g2) and also commute with all the operators of Rv(5), and thus change neither (msm4msm2)nor (FJMso~).The transformation from the Gel'land basis to the one that diagonalises L2 and Lo is of course easy to find numerically. An analytic form for such a transformation that at least diagonalises Lo has been found - for some special cases - by Pang 22). The basis states are now labelled
(£2LNFJMsa),
(24)
where F = 0, 1. . . . f2 and, for each value of F: L =
0, 2 . . . . f 2 - F 1, 3, (2-F
if g2" F even if O - F odd.
Also N = 2Mz+2f2 with Mr. = - L , - L + 1. . . . L. The j-content of the Rr(5 ) representation (F0) is given, e.g. by Wybourne 2z), table C-16. For the few lowest values of F we have F=0: F=I:
J=0, J=2,
F=2: F=3: F=4:
J=2,4, J=0,3,4,6, J = 2, 4, 5,6, 8.
(25)
132
J. K R U M L I N D E
Of course, M s = - J, - J + l . . . . J. The number ~ is an additional quantum number, needed when the same J-multiplet occurs more than once for the same value of F. This is the case only when F > 6. 3.3. T H E H A M I L T O N I A N
Within the basis (t2LNFJMsoO we get the following expressions for the R(8) bilinear Casimir operator: 8
C,(s) = Z J ~ a = f2(82+6);
(26)
1
for the Rv(5) bilinear Casimir operator: 8
CRF(S) = E J2a,
eigenvalue F ( F + 3);
(27)
4
and for the RL(3) Casimir operator 3
CRL(a) = L2 = Z j 2 ,
eigenvalue L ( L + 1).
(28)
1
The monopole pairing interaction takes the form (el. eq. (11)): Vp = - Go(J~23+ Ji3) = - Go( L2 - L20 + Lo).
(29)
The matrix elements of dl 3 and d23 within the Gel'land basis can be found from ref. 21) [see also ref. 11), eqs. (3.38), (39)]. It is now easy to transform to the basis (24) and now CR(S), CRy(5), CRa3) and Vp (el. eqs. (26)-(29)) are all diagonal. The expression for the quadrupole pairing interaction is VvQ = ---~G z { CR( s) -- CRL(3)-- CR(6)},
(30)
where 8
c. 6, = E j2
(31)
3
is not diagonal in the basis (24). Further, for th e quadrupole particle-hole interaction we have 1 VQ = --yz{CR(6)-CRF(5)}. (32) Note that CR(6) commutes with Lo but not with L2. Now, the full Hamiltonian, H = Vp--~-VpQ-~-VQ,
(33)
can be diagonalised numerically, for fixed values of ~2, N and F. Notice that the matrix elements of the Hamiltonian (33) do not depend on J, M s or ~ because of Schur's lemma ( H is invariant with respect to Rv(5)). This procedure of a two-step diagonalisation is useful in the way that the numberof-particles operator is diagonal in the intermediate basis (24). Since we only want one specific value of N in the second step, and since F need not be varied in the first
SCHEMATIC INTERACTIONS
133
step, this method saves, time and space in the computer as well as it yields more physical insight into the problem. 3.4. N U M E R I C A L RESULTS
As is well known, the pure quadrupole interaction gives an ideal rotational spectrum ref. 24); in this case, however, because the full R(8) was taken into account and because of the symplectic invariance, the lowest state has F = 0, I = 0, the next lowest F = 1, I = 2, the third state F = 2, I = 2 and 4, and so on (cf. eq. (25)). The energy relation is E~ = c2F(2F+ 1),
or
E, = cX(I + 1)0
if I denotes the maximum angular momentum for a given value of F (yrast state). A pure monopole pairing interaction gives the ideal seniority spectrum, ~, = - c ( ~ = x - 0 ( I ~ - X + l ) , where again I is the maximum angular momentum in the Rp(5) multiplet (F0) and I=~, = t'a. Fig. 1 shows the spectrum for g2 = 4 with only monopole pairing and quadrupole pairing present (X = 0, Go = 0.1, Gz varied). At Gz = 0 we have something that looks like the phonon spectrum of quadrupole vibrations, only the spacing is not constant (it is the spacing of the seniority spectrum). This is because the additional degeneracy present at G2 = Z = 0 makes the energy of some RF(5) representations equal, thus bunching them in the same way as in the representations of the U(5) of the bosons of the quadrupole vibrations. Also at Go = G2 there is a still higher symmetry due to invariance of the Hamiltonian with respect to the R(6) = {J~0; ~, fl = 3, 4, 5, 6, 7, 8}. The influence of quadrupole pairing is seen very clearly since, already at small values of Gz, the ground state is mixed with the higher 0 + states thus acquiring a stable quadrupole pairing deformation (quadrupole superfluidity). Fig. 2 shows the same situation, but for a large value of X, the coupling constant of the quadrupole particle-hole interaction. Here it is seen especially clearly how the 0 + state specific to quadrupole pairing (i.e. the quadrupole superfluid state) goes down steeply with increasing G2. For this large value of Z, this state acts as an intruder state, having rather small interactions with the normal (not quadrupole-superfluid) states. The quadrupole pairing phase transition of the ground state is delayed to larger values of G 2 when Z is larger. This can also be seen as a compensation phenomenon: the particle-hole and pairing components of the quadrupole tensor counteract each other to some extent; if Gz = Z the sum of them commutes with pairing quasispin, is diagonal in the basis (24) and does not display any quadrupole-type properties. Fig. 3 shows the spectrum as a function of Z with fixed values of G o and G a. The ground state becomes deformed as Z increases but the deformation phase transition is a fairly slow process for realistic and moderately large values of Gz.
134
J. KRUMLINDE J
i
[
I
1
i. ~ - 2 . 4
~0.3.4,B
i ~'F
\
,~,-. 0
\ \'L
:i
o
0
.05
.~
.15 G2
.2
Fig. I. Spectrum o f the multi-two-level model, ~2 = 4, for fixed values o f Go = 0.1, Z = 0, while
G2 is varied.The numbersat the end of the lines denotethe angularmomentaof the states.
-1 -2 -3
uJ
-5 -6
-?
-8
0
.1
.2
.3 G2
~
.5
Fig. 2. See caption to fig. 1. Here Go = 0.1, Z = 0.2, while G2 is varied.
SCHEMATIC INTERACTIONS
135
0 " .'~÷~÷÷+++ -1
"'-
-2 -
*~****+****~ ~:*%
*%
2,4,5,6,8 2.4
"o.
0 -3 0,3,4,6
m
2 -5 2,4 .6
0
-7
2
-8 0
I
r
I
~
.05
.1
.15
.2
0
X Fig. 3. See c a p t i o n t o fig. 1. H e r e Go = 0.1, G2 = 0.05, w h i l e Z is varied.
Generally one finds the usual collective features: monopole and quadrupole pairing vibrations and quadrupole particle-hole vibrations exist, for small values of the relevant coupling constants x = Go, G2, Z; they are high in energy but come down for increasing values of x. Finally they interact with the ground state, thus making the ground state superfluid, quadrupole superfluid and quadrupole deformed, respectively. The suddenness of the phase transitions depend on the interaction matrix elements, which in turn depend on the magnitudes of the other coupling constants. 4. Models with a single-particle splitting 4.11 D I S C U S S I O N O F T H E M O D E L S
The degenerate model explored in sect. 3 has some shortcomings when we try to simulate both spherical nuclei and deformed nuclei. The introduction of a singleparticle energy gap would improve the situation somewhat since we can then compare collective properties with single-particle properties. Without a single-particle gap all effects are collective. The calculations are, however, considerably more complicated, since we have to investigate a model with two different sets of j-shells, in general f21 shells withj = Jr, and at a single-particle energy 2s apart, f22 shells with j = Jz (we may take f21 = ~22 = f2). The full Lie algebra is R(4(jt +jz)+4), but if we do not allow matrix dements of the Hamiltonian to connect states between different sets of shells, the
136
J. KRUMLINDE
problem reduces to that of investigating R(4jl + 2) x R(4j2 + 2). The tensor sequence is R ( 4 ( j l + j 2 ) + 4 ) = { ( T l ° + T ° l + T 1 2 + . . . +T°'2n)} +{(TI°+T°I+T~2+...
+ T°'zJ2)}
(within j l ) (within j2)
+ {T o, IJ~-J~l + T 1, lJl-J~l+ . . . + TO, n+J2+ T 1, ~1+J2} (for intra-shell transitions).
(34)
For example, two sets o f j = -~ shells give a sequence TOO+ 3TlO + 3TOl + Tll + TO2 + 3TlZ + 3T°3 + T 13, and A = ~ and J2 = ½ gives 2T1 o + 3T ol + T 11 + T o2 +27 qz -kZ °3. It may be difficult to extract useful physical information in such a complex situation. Further, the computational work may be prohibitively cumbersome. We shall therefore make a few simplifications and then discuss the physical implications of them. Returning to the degenerate model of sects. 2 and 3 we notice that the T °3 tensor is not explored, but it is necessary to keep it since it occurs in the commutator between different components of the T 12 tensor (cf. eq. (4)). A possible way to remove the Z °3 tensor from the picture is to exchange j for orbital angular momentum !. Then we can reinterpret the two-level model so that the single-particle levels are eigenstates of Iz and spin st, corresponding to the four possible combinations of rn~ = ± 1, m ~ - - +½. It is natural to extend this to a three-level model, including mt = 0(ms = ±½) states also. This completes the N = 1 oscillator shell. The model is equivalent to the previously discussed (j = ~) + (j = ½) model, except that now Iz and s~ are diagonal instead o f t . The tensor sequence is different when we use l as angular momentum viz.
3TOO + TlO + TOl + 3Tll + 3T°2 + T 12. Actually, the spin operators commute with both the orbital angular momentum operators and the pairing quasispin operators. We have for the spin operators: S+ = N 1 3 - N 4 2 + N 5 6 ,
S_ = N a l - N E 4 W N65,
(35)
Sz = ½(N11 -- N22 - N33 + N44 + N55 " N 6 6 ) , b
and for the orbital angular momentum operators t+ = ~ f i ( N ~ + N~4-- N~2 ~36), l_ = x / 2 ( N 5 1 + N , , s - N 2 6 - N 6 3 ) , -
l~ = N l l - N 2 a + N a 3 - N 4 , ~ .
'
(36)
SCHEMATIC INTERACTIONS
137
Tensors with respect to the subgroup RL(3)x Rl(3)x Rs(3) may be denoted T~k~, where (2/0 expresses tensor properties with respect to RL(3), (k~) with respect to R,(3) and (trh) with respect to R~(3). The tensor sequence of R(12) is then R(12) = {T TM + T ° I ° + T * ° ° + T * n
+ T °zl +T*2°}.
The tensor T 12° contains E2 type quadrupole particle-hole and pairing operators, T 1°° is the pairing quasispin, T °*° is orbital angular momentum and T TM is spin. The tensor T TM contains M2 type operators and T 1.1 E1 and M1 operators. We are now mostly interested in T 12°, T 1°° and T °*°. These operators form a subgroup of R(12) which can be seen easily from commutation relations obtained by generalising eq. (4) to include spin. This subgroup is a symplectic group Sp(6) as has been shown by Pomorski and Szpikowski 2s). We denote it SppQ(6) since it contains pairing and quadrupole operators. These operators of SpeQ(6) commute with the spin operators, which follows from the fact that all T zk'r tensors of SppQ(6) have a = 0. In the same way we may form an Sp(6) algebra out of T °21, T °1 o and T TM which commute with paMng quasispin, and thus equals the Flowers Spe(6), cf. sect. 2. Taking only # = 0 components (i.e. only single-particle operators N~p) of the tensors T 12°, T 1°°, T °I° reduces Sppo(6 ) to SUpo(3), the wellknown SU(3) group of Elliott 24). Still, the group Sp(6) is cumbersome to work with and an additional simplification will be introduced. We may consider only operators of a two-dimensional N = 1 oscillator shell, in which case the threedevel model is again reduced to only two levels; we retain the levels of m t = ± 1, m s = ±½ and just the operators that connect these levels. Then R(12) is reduced to R(8). The group of pairing plus quadrupole operators SppQ(6) is reduced to SppQ(4)--~ RpQ(5), containing the T ~°° tensor, the q = 0 component of T °1° (viz. the operator lo) and the q = ___2 components of T 12°. This R(5) group has previously been explored in some detail, cf. refs. 26-28). The Elliott SU(3) reduces in the same way to an SU(2) or R(3), previously discussed by Lipkin 29). The group R(5) is sufficiently simple as to allow for a generalisafion into RpQ(5) x RpQ(5), corresponding to two sets of shells with a single-particle splitting. This problem has also been considered previously 3o) and we refer to refs. 28,ao) for the method of solution. Our Hamiltonian is now H = ~(N(1) - N(2)) - Go(L +(1) + L + (2))(/,_ (1) + L_ (2)) -- ~Gz(T,t22(1) + T~22(2))(T,*22(1)+ T,az2(2))t -kG2(T)-22(1) + 7",1-22(2))(7"11_22(1) + T?_22(2)) t
- l z ( T g 0 ) + T g ( 2 ) ) ( r g 0 ) + TO#(2))' --~Z(Tol-Z2(1) + Td2.-z(2))(To*Z_2(1) + T~_22(2))t,
(37)
138
J. K R U M L I N D E
where arguments (1) and (2) refer to upper and lower sets of shells, respectively. In the notation of ref. no) (cf. also ref. 2s)) we have
n = KNO)- N(2))- Oo v,-
zvQ
-~G2{(Vt++ Uz+)(Ua- + Uz-)+(V, ++ Vz+)(V1- + V2-)}. (38) 4,2. N U M E R I C A L
RESULTS
With the model described above we may now test the collectivity of states with respect to the various degrees of freedom corresponding to the different interactions and also the competition with the single-particle effects. The magnitudes of the coupling constants are not directly comparable with those in sect. 3 since the full rotationally invariant quadrupole type interactions were not used here. Fig. 4 shows the energy difference between the first excited 0 + state and the ground state (upper half) and the excitation energy of the lowest 2 + state (lower half) as a function of the single-partMe energy gap ~. The leftmost part of the figure shows the behaviour with only monopole pairing present (Go = 0.1, G2 = Z = 0). The state shows no decidedly collective properties at pair degeneracy D = 1 but the collectivity increases with D, i.e. the excitation energy becomes small within a certain interval o f the parameter ~. For D = 3 the state has a pairing vibrational character around
D=I
.Q=I .
.Q=I
i
to T D=3
,
,
,
t
I
0
>
.5
.5
i
ID=I /~Q=2 1
1
E2T
E2T
~ , , ' ~ S 2
i
i
i
I .5
•
0
~
~
~
=3
I
i
I
l
•
.5
[
0 I
0
.5
e
.Q=I 1
~2T~
.5
0
]
0
E2;
3
~
~
f
~
I
.5
~
i
0
0
,
:
I
I .5
Fig. 4. T h e excitation energy of the first excited 0 + state (upper part) and the lowest 2 + state (lower part) in the model described in subsect. 4.1. T h e p a r a m e t e r e is varied. The leftmost part o f the figure shows the results for Go = 0.1, Gz = Z = 0, the left central part Go = 0.1, G2 = 0.4, Z = 0, the right central part Go = 0.1, G2 = 0, Z = 0.1, a n d the rightmost part Go = O.1, G2 = 0.4, Z = 0.1.
SCHEMATIC INTERACTIONS
139
8 = 0.4. This phenomenon was previously investigated by Hoegaasen-Feldman 1) within (for this special case) essentially the same model and with the same qualitative result. The 2 + state, shown in the lower leftmost of the figure, shows no collectivity, since this state is not specifically influenced by the monopole pairing. The specific parameter determining the energy of this state is Go/Gerit where aerit ---- /~/~'~, thUS giving an increase of excitation energy with increasing e, with smaller slope if f2 is larger. The left central part of the figure shows the behaviour of these states with both monopole pairing (Go = 0.1) and quadrupole pairing (G 2 = 0.4) present but no quadrupole particle-hole interaction (X = 0). The 0 + state displays the same behaviour as with only monopole pairing present, although here the collectivity is essentially due to the quadrupole pairing interaction. The 2 + state (left central, lower) shows a less transparent pattern: for f2 = 1 there is no collectivity, for f2 = 2 this state manifests itself as a well-developed quadrupole-pairing vibration, for f2 = 3 the ground state is quadrupole superfluid and the nucleus is again less soft against changes in the coupling constants. The right central part shows the states with monopole pairing (Go = 0.1) I and quadrupole particle-hole interaction (Z = 0.1) but no quadrupole pairing. The ground state is now deformed. The first excited 0 + state (upper part) shows some collectivity at f2 = 1 (beta vibration) b u t a t f2 = 2 and 3 the interference of the monopole pairing degree of freedom washes out the structure and no information about the character of the states can be inferred from just looking at the energy. The energy of the 2 + state (lower part) increases with z, reflecting the fact that the moment of inertia decreases with increasing energy gap. The rightmost part of fig. 4 shows the energy of the first excited 0 + state and the lowest 2 + state with all three interactions present (Go = 0.1, G2 = 0.4, Z = 0.1). The interference of the various collective modes makes it diificult to draw any conclusions from looking at only energies. The 2 + states show some collectivity of quadrupole pairing and quadrupole particle-hole character for f2 > 2. The destructive interference between the pairing and particle-hole components of the quadrupole interaction makes the ground state less deformed and more soft against changes in the coupling constants than in the case of only monopole pairing and quadrupole particle-hole interactions. Thus the 2 + state shown in the lower rightmost part of fig. 4 is not of rotational character. Fig. 5 tests the sensitivity of the ground state and the first excited 0 + state to changes in the coupling constants, looking at one at a time. The upper left part shows the variation of the excitation energy with the strength of the quadrupole pairing interaction, for fixed values of Go = 0.1, Z = 0.1 and f2 = 2. Two different values of e are employed: e = 0 and ~ = 0.2. The curves show very clearly that there is a region of (72 for which the state is a low-lying collective vibration; this value of Gz is of course shifted to a larger value when ~ is larger. The lower left part shows the variation of the expectation value of the quadrupole pairing interaction itself, thus
140
J. K R U M L 1 N D E
+-
¥
tu
~=0.2
~=0.2
i .2
a
i
i
J+
.6
.8
0 0
i
i
s
1
.05
.I X
.15
.2
Gz
8
16
i ',,t-',/ ~,
~
,-
Q
.2
.0
.8
J
e=t~2
2O
0
.OS
.1 X
/ ~ ' = 0
,-, /
i t' "J \i t/ ( '7., ,t / // ~/ i t , ) .>({__.__/'..=:o5
....
.t, Gz
0 G
/ f " "
/--\
•
.15
%
<:o.2
.2
G
.1
Fig. 5. Excitation energy of the first excited 0 + state (tipper part) in the model described in subsect. 4.1. Here g2 = 2 and curves for a few different values o f e are shown. The leftmost part shows results for fixed values of Go = 0.1, Z = 0.1, while G2 is varied; the central part Go = 0.1, Gz = 0.2, while Z is varied and the rightmost part Gz = 0,2, Z = 0.05, while Go is varied. The lower part o f the figure gives the expectation values of the relevant interaction, thus showing the real correlation effects. In the leftmost part the quadrupole pairing expectation values are shown, in the central part quadrupole particle-hole, and in the rightmost part monopole pairing expectation values. Solid lines show the behaviour of these quantities for the ground state, dashed lines for the first excited 0 + state.
1.0
t.O
.8
.8
~.~ t~ J .2
0
0
.I
.2
~3
02
.4
.5
.I
.2
.3
1
I
,¢
.5
I
G2
Fig. 6. Ratio of the (t, p) and (p, t) cross section from the ground state in the n-particle system to the first excited 0 + and the ground state in the (n~2)-particle system. Here ,Q = 2, n = 4. Go = 0.1, e = 0.4, while G2 is varied. The left part has Z = 0.I, the right part Z = 0.2.
SCHEMATIC INTERACTIONS
141
testing specifically the collectivity for this particular mode. It is seen that the quadrupole superfluidity of the ground state (solid line) increases conspicuously just in the region where the first excited 0 + state has the collective character, indicating that these features are the results of a phase transition. The qnadrupole superfluidity of the excited state (dashed line) shows a slightly more complicated behaviour: it first increases when G2 increases but after assuming a maximum it decreases (for e = 0), thus leaving the larger part of the interaction strength to the ground state. At e = 0.2 this feature is less clearly displayed as the presence of the single.particle field to some extent inhibits and smoothes out the collective effects. The central part of the figure shows the same results for the quadrnpole particlehole interactions (here Go = 0.1, G2 = 0.2, Z is varied). The spherical-deformed phase transition is very clearly borne out also for rather large values of 8; the first excited state is first spherical, then of a quadrupole or beta vibrational character in the phase transition region and finally again less collective, because a large part of the strength has now gone into the ground state. The rightmost part of fig. 5 shows the results of variation of the monopole pairing coupling constant (here G2 = 0.2, Z = 0.05) for a nearly spherical system. The presence of a strong quadrupole pairing field interferes seriously with the monopole pairing and smoothes out the phase transition. Finally fig. 6 shows the (t, p) and (p, t) cross sections, displaying the asymmetry resulting from a strong quadrupole pairing interaction. Because of the different single-particle quadrupole moments of the mj = ___)levels and the m~ = -t-½ levels the pair addition and pair removal modes in a model without particle-hole symmetry (here f2 = 2, n = 4) will have different transition rates, since monopole pairing and quadrupole pairing may interfere destructively in the one mode and constructively in the other mode. This phenomenon is discussed more elaborately by B~s et aL 6) and the result of this solvable model seem to confirm their physical picture. In conclusion one may say that the various phase transitions (normal-superfluid, quadrupole normal-quadrupole supeffluid, spherical-quadrupole deformed) are more clean and usually also more sudden if the corresponding interaction is more pure. Then the vibrations can be extremely collective at certain points. The presence of other interactions smoothes out these features, just like the presence of a single-particle field does. Generally, if many interactions are present there is not much information to gain from just looking at the energies since the interference between the different modes of collectivity may sometimes make the energy of the collective state insensitive to changes in the coupling constants. In such cases the state may change considerably the character of the collectivity without showing this in the energy. Then it is essential to look at specific quantities related to the correlations of the interactions themselves, like two-particle transfer cross sections and multipole transition probabilities. The author is very much indebted to Ricardo A. Broglia for many inspiring and exciting discussions, which finally made this work possible.
142
J. K R U M L I N D E
References 1) J. Hoegaasen-Feldman, Nucl. Phys. 28 (1961) 258 2) A. K. Kerman, Ann. of Phys. 12 (1961) 300 3) D. R. B~s and R. A. Broglia, Nucl. Phys. 80 (1966) 289; A. Land6, Ann. of Phys. 31 (1965) 525; A. Pawlikowski and W. Rybarska, JETP (Soy. Phys.) 16 (1963) 38; R. W. Richardson and N. Sherman, Nucl. Phys. 52 (1964) 221; J. Bang and J. Krumlinde, Nucl. Plays. A141 (1970) 18 4) A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 30 (1955) no. 1 5) J. P. Elliott, Proc. Roy. Soc. A245 (1958) 128; A245 (1958) 562 6) D. R. B~s, R. A. Broglia and B. Nilsson, Phys. Lett. 40B (1972) 338; 50B (1974) 213 7) R. A. Broglia and I. Ragnarsson, to be published; R. A. Broglia and B. Nilsson, to be published 8) L Hamamoto, preprint, Nucl. Phys. A232 (1974) 445 9) K. Hara, Proc. 7th summer school, Masurian Lakes, 1974, to be published in Nukteonika 10) J. Krumlinde and Z. Szymafiski, Phys. Lett. 53B (1974) 322 11) J. Krumlinde and Z. Szymafiski, Ann. of Phys. 79 (1973) 201 12) J. Krtmalinde and Z. Szymafiski, Nucl. Phys. A221 (1974) 93 13) B. H. Flowers, Proc. Roy. Soc. A212 (1952)248 14) A. R. Edmonds and B. R. Flowers, Proc. Roy. Soc. A214 (1952) 515 15) K. Helmets, Nucl. Phys. 12 (1959) 647 16) K. Helmers, Nucl. Phys. 23 (1961) 594 17) S. T. Belyaev, Some aspects of collective properties in nuclei, in Selected topics in nuclear theory, ed. F. Janouch (IAEA, Vienna, 1963); Nucl. Phys. 24 (1961) 322 18) A. B. Migdal, JETP (Soy. Phys.) 37 (1959) 249; Nucl. Phys. 13 (1959) 655 19) D, E. Littlewood, The theory of group characters (Oxford University Press, Oxford, 1950) p. 299 20) B. H. Flowers and S. Szpikowski, Proc. Phys. Soc. 84 (1964) 673 21) I. M. Gel'land and M. L. Cetlin, Dokl. Akad. Nauk SSSR 71 (1950) 1017 22) S. C. Pang, Nucl. Phys. A128 (1969) 497 23) B. Wybourne, Symmetry principles and atomic spectroscopy (Wiley, New York, 1970) 24) J. P. Elliott, The nuclear shell model and its relation with other models, in Selected topics in nuclear theory, ed. F. Janouch (IAEA, Vienna, 1963) 25) K. Pomorski and S. Szpikowski, Acta Phys. Pol. B1 (1970) 3 26) P. Chattopadhyay, F. Krejs and A. Klein, Phys. Lett. 42B (1972) 315 27) C. Dasso, F. Krejs, A. Klein anti P. Chattopadhyay, Nucl. Phys. A210 (1973) 429 28) J. Krumlinde and E. R. Marshalek, Nuovo Cim. Lett. 7 (1973) 679 29) H. J. Lipkin, Nuel. Phys. 26 (1961) 147 30) J. Krumlinde and A; Kaijser, Nuovo Cim. Lett. 8 (1973) 669
[
Nuclear Physics A243 (1975) 125--142; @) North-ltollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON SCHEMATIC INTERACTIONS IN SOLVABLE MODELS, ESPECIALLY THE QUADRUPOLE PAIRING INTERACTION J. K R U M L I N D E Department of Mathematical Physics, Lund Institute of Technology, S-220 07 Lund 7, Sweden Received 28 November 1974 Abstract: TJ~e properties of and the interplay between the monopole pairing, cluadrupole pairing
and quadrupole particle-hole interactions are studied in two simple, solvable models. i. Introduction The study of schematic interactions and the exploration of their properties by means of exact solutions in solvable models plays an important role in the understanding of nucleonic correlation effects in medium and heavy nuclei. Especially simple and useful interactions are, for instance, the ordinary (monopole) pairing interaction and the ordinary (single-particle or particle-hole type) quadrupolequadrupole interaction. These have been studied extensively previously l - s ) and their properties are well established throughout the whole range of values of the coupfing constants. The quadrupole pairing interaction is an interesting hybrid of pairing and quadrupole as it introduces correlations of quadrupole type in the particle-particle (pairing) component of the nuclear two-body interaction. As has been demonstrated recently [refs. 6-8)] it can offer an explanation of effects which were difficult to describe with only ordinary pairing and quadrupole interactions, such as the state-dependent pairing gap, introducing asymmetry between (t, p) and (p, t) cross sections 6, 7), and attenuation of Coriofis matrix elements in rotational states s, 9). This paper will try to describe the properties of the quadrupole pairing interaction and to investigate the effects of the interplay of the three schematic interactions. We also show how these interactions emerge in a natural way and thus yield the basic properties of nuclear interactions in some very simple solvable models. The influence of the quadrupole pairing interaction on rotational properties wilL however, not be discussed here as it is the subject of a separate paper 10). 2. Interactions in a multi-j-shell model Consider an n-level model, each level having, except for the twofold degeneracy due to time,reversal invariance of the Hamiltonian, an additional pair degeneracy ~2. 125
126
J. K R U M L I N D E
We introduce creation and annihilation operators for the levels ~ -- 1 . . . 2n, v = 1 . . . f2: eva, t cva. Further, we define single-particle operators, pair creation operators and pair annihilation operators: t c V=I
o
"t c~a,
(1)
V=l
see also refs. 11,12). The levels are labelled so that ~ = 2 n - l(m~ > 0) and ~ = 2n(m~, < 0) are time+ = J ' c ~+, 2 , - i ~ - --I . The operators defined in eq. (1) are the reversed partners, c~,2, generators of the rotation group R(4n). The transformation from the operators (1) to the R(4n) angular momenta L,p is given, for instance, in ref. 12), eqs. (44) and (45). If we assume that our model consists of f2 identical j-shells, the physical angular momentum is given by, assuming a basis withj~ diagonal: j_ = ~ x / ( i - 1 ) ( 2 n - i + 1)(N2~_ i , 2 i - 3 - N2i-2,2i)+ nN2n,2,-1, i=2 /1
j+ =j*--,
= i=l
n = j+½.
(2)
For the classification of the states of the n-level model the operators of the total pairing quasispin are of great systematic importance: L+ =
t=l
B'2,-1,2,,
L , = L*+,
(3)
/I
Lo = ½ Z (N2,-i,2,-I + N21,2i)-½nl2" f=l
Pairing quasispin and physical angular momentum form two commuting SU(2) algebras, together forming an R(4) algebra RLi(4). Thus R(4n) = RLj(4), and the operators of R(4n) can be classified as tensors with respect to the group
SCHEMATIC INTERACTIONS
127
RLj(4). We definetensor operators T~ in the followingway: [L+, T~] = (2(2 + 1 ) - #(/l +_1))'}T~ a,q,
ELo, Tu~] U_+,
=
#T~,
= (k(k + 1 ) - q(q_+
(4)
= qr¢ . The tensor operators fulfill the following conditions of hermiticity and time reversal: ( "~,~+qT T ~ t = ,.--., - - 2k" ~,-q,
~ - T ~ ---1 = (--)~+aT~a.
(5)
The commutation relations are of the form:
[T¢:::, r¢:2] = C(2, 2~ k~ kz) Z (2~ 2,- ~ ~ +~212~ ~ 22 ~2> Ak
x
(6)
where C(2x22k~k2) is connected with the reduced matrix elements of the tensors T ~k. Eq. (6) is useful as it provides us with certain selection rules. It is easily seen that the operators defined by eq. (1) can be classified as RLj(4) tensors in the following way:
= {TlO +T°l-}-T~2+ TO3+ . . . + T 1,2i- l + TO,2i}. (v) Here T x° is the pairing quasispin: {N~,B~,B~p}~,~=l...2,,=21+l
7"+1°o = T-x/~-L_+,
Toi° = Lo,
(8)
TOo~ = j~.
(9)
and T °i is the physical angular momentum O1 T~+,
T~/~j+,
Tensors T ~k with 2 -- 1 will be called quasivectors, those with 2 = 0 quasiscalars. The quantum number k(or rather 2 ) is called the multipole order. A unitary subgroup Uo(2n) is formed by all # = 0 components of the T ~ tensors, i.e. all single-particle operators N~. Another subgroup of R(4n) - and also a subgroup of U0(2n) - can be formed out of all quasiscalar tensor components T o, z,-,. This is the symplectic group as employed by Flowers l~, ~ ) and others l s , ~ ) . It is here referred to as Sp~(2n). The operators belonging to this subalgebra commute with the total pairing quasispin operators, which is easily seen, e.g., from eq. (4). Thus, the pairing quasispin operators are symplectic invariants (invariant by transformations generated by Spr(2n ) generators) and, likewise, all Sp~(2n) operators are invariant by Bogolyubov-Valatin transformations, el. e.g. ref. 1s). The most general nuclear two-body interaction of the multi-j-shell model is composed of bilinear expansions in T~, forming rotationally invariant, hermitian combi-
128
J. K R U M L I N D E
nations of the the type type nations of '=°
= E
(_Y)"
v -"q -gq " = ~/2k+l ~
(10)
Taking 2 = 1, k = 0 gives the (ordinary) monopole pairing interaction, 2 = 1, = 0, k = 2 gives the quadrupole particle-hole interaction and 2 = 1,/~ = 1, lc = 2 the quadrupole pairing interaction. Actually, it is easily verified that t
j
Zff =
E
m,m'=-j
t
(ajmajm,,
¢t = I
{a~majm,, ~ .ajmajrn, ,
I.Z = 0 lot = - - 1 ,
using creation and annihilation operators labelled with the (j, m) quantum numbers. The gauge invariant pairing interaction 17. ~s) or surface delta interaction is given by 6 = E (2" 2k+ 1)-l[T)-=~kT#-:_~,]Jo=°. (12) k This is in general not a symplectic invariant 1s), since only T 1o commutes with all generators of Spe(2n). An exception from this rule is the multi-two-level model (j = ~) discussed below. The general form of the two-body interaction in this model is thus
V = 2 Z1,2 To°kaTo°~*+ 2 kodd
q
~.
Zk., 2 T1aakT~*:
keven ~t=--l, O, 1
(13)
q
If, further, Zk~is independent of/~ there is an additional degeneracy because V then commutes with L2, the total pairing quasispin. This condition is not fulfilled in the realistic cases, but such solutions provide us with interesting limiting cases which can be useful in the classification of states. 3. The multi-two-level model 3.1. I N T E R A C T I O N S
As is discussed, e.g. in Littlewood 19), the group R(8) possesses a group of six outer automorphisms, isomorphic with the symmetric group $3, interchanging irreducible R(8) representations of the same dimensions, cf. also Flowers and Szpikowski 2o). Thus it is possible to perform a non-trivial, unitary and canonical transformation of the generators L~a onto new generators J ~ , which are just as good as the old ones and more convenient for oar purpose. Note that this is possible only for R(8). See also ref. 12), subsect. 4.3, w ~ e such a transformation is given. In this new scheme the number-of-particles 0~6rator equals one of the generators, which turns out to be a great advantage. For convenience, table 1 gives the relations between the tensor operators, defined in sect. 2, and the g(8) generators, both in terms of the operators N~p, B~p, B~p and in terms of the specific R(8) operators r~. The direct relation between N~p, etc., and J~p is given, e.g. in ref. 11), table 1.
SCHEMATIC INTERACTIONS
129
TABLE 1 Definition of tensor operators
= ½ 4 3 ( J ~ 6 + s~ ~ ) - i K / 3 ( J ~ s + J67)
=
= -4~(s,+u,o) = -s~ is,7 + + 4 ~ ( s ~ - s ~ . )
=
x/3Ni2 ~/-~(i~+N**)
-
= N4z -- N~ 3 + ~/3Nz*
-~½43(A~-A~)
To°2
= ½(N , ~- N z ~ ) - + ( N ~ -
= 2Js7 +J6s
O3
N,,)
= J,5 - iJ47 - ½~/3(J56 - J 7 8 ) - i ½ 4 3 ( J 5 8 - J 6 7 ) = N31 - N v , -
To°32
= -4~(J48
= 4 ~ ( 1 % + N,~)
-/J,6)
7-o0_33
=
T~t~l
---- B~3
= - ½ ( J ~ - J 2 7 ) - i½(J25 + J xT)
---- B ~ , :T
=
TL~2
--45N2t
= ½(J26 - J 1 8 ) - - i½(S, 6 + Su8)
=
-4~(J2,-iJ14) --
~/3~43
t
= -4~W~-BL)
½(St 5 + J z T ) - i½(J2~
- -
= B~3
S17)
= +(s,~ + s ~ ) - ~ ½ ( s ~ - s ~ )
---- B~4
= - 4 ~T( J ~ 6 - * J ~ . )
= -~/T(sa7-iSa~ )
= - 4 ~ ( N , ~ + N~_0
J34
:/-oi_2, T~22
= ,
= ½(N11 + N22 - Na 3 - N44)
x[T(s37+iS~5) -~/~(J~6+U~.)
= 4~(N~, + N~) = 4 ~T( ,N, - N ~ 3 )
- ½ ( s ~ + s ~~ ) - ~½(s ~~ - J~.) 12 T-*it
= --824
= -- ½(Ja5 -q--J 2 7 ) + i½(J2s -- Sl 7)
= B23
12
= 4~(8I~-B3,)
7"1_~_ ~ = - ½ ( j ~ - z . ) + i½(A~ + z ~7)
= B14
12
T:~ _ ~ = - ~(S: ~ - J~ ~ ) - ~½(Z~~ + A ~) ToOl
= --813
i= - 4 ~T( J 5 6 - J7 s) - x/~(S67 - J58)
- 4~(J~ + iJ~7) -- -4~(s~-
N,~)-4~S~
ro°o~ T
T:oo To%o
=
= J12
10
T_~lo
4Ta_(j13+iJ23)
=
-4-~(J~3-iJ23)
T
+ 4 ~ ( J , ~ - ~J,~) = 4 ~ ( N ~ - N~,) + 4 ~ N , ~ _ -4~(B~2+B34) T ~ "t _
= ½(Nl~+N22+N33+N44)-f2
= 4~(B~2+B34)
130
J. K R U M L I N D E
The monopole pairing interaction takes the form Vz, = --Go ~rlO-r!Ot ~,o~,o = - - G o ( J 123 + J 22a ) ,
(14)
with Go -- Zo, the quadrupole pairing interaction VpQ =
--+GzZ
8
Ti12T:• "t =
-~G2 Z
2 -1-J2/I), 2 (J1/3
(15)
#=4
q
1 with -~G2 = )~z.1, and the quadrupole particle-hole interaction 8
VQ = ---~;g Z Tol~Tol~¢r = --~Z E S~#, q
(16)
.8=4
with @Z = Z2,o. The Flowers symplectic group Sp~(4) is generated by the three operators of T °I (physical angular momentum) and the seven operators of T °a (the quasiscalar octupole tensor). Its Lie algebra is isomorphic to the Lie algebra of a certain R(5) group, namely SpF(4) -~ RF(5) = {J~a; a, fl = 4, 5, 6, 7, 8}, (17) It is easily verified that the interactions Vp, Vt,Q and VQ, defined in eqs. (14), (15) and (16) respectively, commute with all the generators of RF(5). Thus all these interactions are sympleetic invariants, in contrast with what is the case for the general mu/ti-n-level model w i t h j ¢ ~. 3.2. T H E G E L ' F A N D BASIS A N D T H E P H Y S I C A L BASIS
The low-lying states span the spaces that carry the symmetric representations, el. ref. 1~), subsect. 3.10. As a basis we use the one defined by Gel'land and Ceflin 22). For the group R(8), such a basis state is, in Gel'land notation im 8 m7 m6 m5 m4 ma km2
0 0 0~ 0 0 0 0 0 , 0 0 0 0
(18)
here just shortened to
(m8 m7 m6 ms m,~ m3 m2).
(18')
The quantum number ms characterizes the representation itself. It is identical to ~2, the pair degeneracy. The dimension of the representation (~2) is given by de = ~6&~(I2+ 5)(~2 + 4)(0 + 3)2(0 + 2)(0 + 1). Now, the R(5) group in the chain R(8) = R(7) = . . .
= R(2),
(19)
SCHEMATIC INTERACTIONS
131
defining the Gel'fand scheme, is taken to be the Flowers symplectic group Spy(4) or RF(5). The quantum number F = ms,
(20)
characterizes the RF(5) representations (F0), the dimension of which is given by dr = I ( F + )(2F+3)(F+2).
(21)
Since RF(5)~Rj(3) where Ri(3) is the physical angular momentum group, we may change the basis from the one characterized by (m4m3m2)to a basis wherej z andj~ are diagonal, thus exchanging (m4mam2)for (JMsa) where ~ is an additional quantum number sometimes needed (see below). Further, due to R(8) ~ RL(3) X R~(5), we can change the basis so as to exchange the quantum numbers (LN), eigenvalues of the pairing quasispin operators L2 and L0.
(22)
(m7m6)for (LML) or
L21LML> = L(L + 1)[LMz>, LolLML) = MzILMr.),
(23)
N = 2Mz+212, where N denotes the number of particles. This is possible since L2 and Lo commute with the R(8) Casimir operators (and thus do not change g2) and also commute with all the operators of Rv(5), and thus change neither (msm4msm2)nor (FJMso~).The transformation from the Gel'land basis to the one that diagonalises L2 and Lo is of course easy to find numerically. An analytic form for such a transformation that at least diagonalises Lo has been found - for some special cases - by Pang 22). The basis states are now labelled
(£2LNFJMsa),
(24)
where F = 0, 1. . . . f2 and, for each value of F: L =
0, 2 . . . . f 2 - F 1, 3, (2-F
if g2" F even if O - F odd.
Also N = 2Mz+2f2 with Mr. = - L , - L + 1. . . . L. The j-content of the Rr(5 ) representation (F0) is given, e.g. by Wybourne 2z), table C-16. For the few lowest values of F we have F=0: F=I:
J=0, J=2,
F=2: F=3: F=4:
J=2,4, J=0,3,4,6, J = 2, 4, 5,6, 8.
(25)
132
J. K R U M L I N D E
Of course, M s = - J, - J + l . . . . J. The number ~ is an additional quantum number, needed when the same J-multiplet occurs more than once for the same value of F. This is the case only when F > 6. 3.3. T H E H A M I L T O N I A N
Within the basis (t2LNFJMsoO we get the following expressions for the R(8) bilinear Casimir operator: 8
C,(s) = Z J ~ a = f2(82+6);
(26)
1
for the Rv(5) bilinear Casimir operator: 8
CRF(S) = E J2a,
eigenvalue F ( F + 3);
(27)
4
and for the RL(3) Casimir operator 3
CRL(a) = L2 = Z j 2 ,
eigenvalue L ( L + 1).
(28)
1
The monopole pairing interaction takes the form (el. eq. (11)): Vp = - Go(J~23+ Ji3) = - Go( L2 - L20 + Lo).
(29)
The matrix elements of dl 3 and d23 within the Gel'land basis can be found from ref. 21) [see also ref. 11), eqs. (3.38), (39)]. It is now easy to transform to the basis (24) and now CR(S), CRy(5), CRa3) and Vp (el. eqs. (26)-(29)) are all diagonal. The expression for the quadrupole pairing interaction is VvQ = ---~G z { CR( s) -- CRL(3)-- CR(6)},
(30)
where 8
c. 6, = E j2
(31)
3
is not diagonal in the basis (24). Further, for th e quadrupole particle-hole interaction we have 1 VQ = --yz{CR(6)-CRF(5)}. (32) Note that CR(6) commutes with Lo but not with L2. Now, the full Hamiltonian, H = Vp--~-VpQ-~-VQ,
(33)
can be diagonalised numerically, for fixed values of ~2, N and F. Notice that the matrix elements of the Hamiltonian (33) do not depend on J, M s or ~ because of Schur's lemma ( H is invariant with respect to Rv(5)). This procedure of a two-step diagonalisation is useful in the way that the numberof-particles operator is diagonal in the intermediate basis (24). Since we only want one specific value of N in the second step, and since F need not be varied in the first
SCHEMATIC INTERACTIONS
133
step, this method saves, time and space in the computer as well as it yields more physical insight into the problem. 3.4. N U M E R I C A L RESULTS
As is well known, the pure quadrupole interaction gives an ideal rotational spectrum ref. 24); in this case, however, because the full R(8) was taken into account and because of the symplectic invariance, the lowest state has F = 0, I = 0, the next lowest F = 1, I = 2, the third state F = 2, I = 2 and 4, and so on (cf. eq. (25)). The energy relation is E~ = c2F(2F+ 1),
or
E, = cX(I + 1)0
if I denotes the maximum angular momentum for a given value of F (yrast state). A pure monopole pairing interaction gives the ideal seniority spectrum, ~, = - c ( ~ = x - 0 ( I ~ - X + l ) , where again I is the maximum angular momentum in the Rp(5) multiplet (F0) and I=~, = t'a. Fig. 1 shows the spectrum for g2 = 4 with only monopole pairing and quadrupole pairing present (X = 0, Go = 0.1, Gz varied). At Gz = 0 we have something that looks like the phonon spectrum of quadrupole vibrations, only the spacing is not constant (it is the spacing of the seniority spectrum). This is because the additional degeneracy present at G2 = Z = 0 makes the energy of some RF(5) representations equal, thus bunching them in the same way as in the representations of the U(5) of the bosons of the quadrupole vibrations. Also at Go = G2 there is a still higher symmetry due to invariance of the Hamiltonian with respect to the R(6) = {J~0; ~, fl = 3, 4, 5, 6, 7, 8}. The influence of quadrupole pairing is seen very clearly since, already at small values of Gz, the ground state is mixed with the higher 0 + states thus acquiring a stable quadrupole pairing deformation (quadrupole superfluidity). Fig. 2 shows the same situation, but for a large value of X, the coupling constant of the quadrupole particle-hole interaction. Here it is seen especially clearly how the 0 + state specific to quadrupole pairing (i.e. the quadrupole superfluid state) goes down steeply with increasing G2. For this large value of Z, this state acts as an intruder state, having rather small interactions with the normal (not quadrupole-superfluid) states. The quadrupole pairing phase transition of the ground state is delayed to larger values of G 2 when Z is larger. This can also be seen as a compensation phenomenon: the particle-hole and pairing components of the quadrupole tensor counteract each other to some extent; if Gz = Z the sum of them commutes with pairing quasispin, is diagonal in the basis (24) and does not display any quadrupole-type properties. Fig. 3 shows the spectrum as a function of Z with fixed values of G o and G a. The ground state becomes deformed as Z increases but the deformation phase transition is a fairly slow process for realistic and moderately large values of Gz.
134
J. KRUMLINDE J
i
[
I
1
i. ~ - 2 . 4
~0.3.4,B
i ~'F
\
,~,-. 0
\ \'L
:i
o
0
.05
.~
.15 G2
.2
Fig. I. Spectrum o f the multi-two-level model, ~2 = 4, for fixed values o f Go = 0.1, Z = 0, while
G2 is varied.The numbersat the end of the lines denotethe angularmomentaof the states.
-1 -2 -3
uJ
-5 -6
-?
-8
0
.1
.2
.3 G2
~
.5
Fig. 2. See caption to fig. 1. Here Go = 0.1, Z = 0.2, while G2 is varied.
SCHEMATIC INTERACTIONS
135
0 " .'~÷~÷÷+++ -1
"'-
-2 -
*~****+****~ ~:*%
*%
2,4,5,6,8 2.4
"o.
0 -3 0,3,4,6
m
2 -5 2,4 .6
0
-7
2
-8 0
I
r
I
~
.05
.1
.15
.2
0
X Fig. 3. See c a p t i o n t o fig. 1. H e r e Go = 0.1, G2 = 0.05, w h i l e Z is varied.
Generally one finds the usual collective features: monopole and quadrupole pairing vibrations and quadrupole particle-hole vibrations exist, for small values of the relevant coupling constants x = Go, G2, Z; they are high in energy but come down for increasing values of x. Finally they interact with the ground state, thus making the ground state superfluid, quadrupole superfluid and quadrupole deformed, respectively. The suddenness of the phase transitions depend on the interaction matrix elements, which in turn depend on the magnitudes of the other coupling constants. 4. Models with a single-particle splitting 4.11 D I S C U S S I O N O F T H E M O D E L S
The degenerate model explored in sect. 3 has some shortcomings when we try to simulate both spherical nuclei and deformed nuclei. The introduction of a singleparticle energy gap would improve the situation somewhat since we can then compare collective properties with single-particle properties. Without a single-particle gap all effects are collective. The calculations are, however, considerably more complicated, since we have to investigate a model with two different sets of j-shells, in general f21 shells withj = Jr, and at a single-particle energy 2s apart, f22 shells with j = Jz (we may take f21 = ~22 = f2). The full Lie algebra is R(4(jt +jz)+4), but if we do not allow matrix dements of the Hamiltonian to connect states between different sets of shells, the
136
J. KRUMLINDE
problem reduces to that of investigating R(4jl + 2) x R(4j2 + 2). The tensor sequence is R ( 4 ( j l + j 2 ) + 4 ) = { ( T l ° + T ° l + T 1 2 + . . . +T°'2n)} +{(TI°+T°I+T~2+...
+ T°'zJ2)}
(within j l ) (within j2)
+ {T o, IJ~-J~l + T 1, lJl-J~l+ . . . + TO, n+J2+ T 1, ~1+J2} (for intra-shell transitions).
(34)
For example, two sets o f j = -~ shells give a sequence TOO+ 3TlO + 3TOl + Tll + TO2 + 3TlZ + 3T°3 + T 13, and A = ~ and J2 = ½ gives 2T1 o + 3T ol + T 11 + T o2 +27 qz -kZ °3. It may be difficult to extract useful physical information in such a complex situation. Further, the computational work may be prohibitively cumbersome. We shall therefore make a few simplifications and then discuss the physical implications of them. Returning to the degenerate model of sects. 2 and 3 we notice that the T °3 tensor is not explored, but it is necessary to keep it since it occurs in the commutator between different components of the T 12 tensor (cf. eq. (4)). A possible way to remove the Z °3 tensor from the picture is to exchange j for orbital angular momentum !. Then we can reinterpret the two-level model so that the single-particle levels are eigenstates of Iz and spin st, corresponding to the four possible combinations of rn~ = ± 1, m ~ - - +½. It is natural to extend this to a three-level model, including mt = 0(ms = ±½) states also. This completes the N = 1 oscillator shell. The model is equivalent to the previously discussed (j = ~) + (j = ½) model, except that now Iz and s~ are diagonal instead o f t . The tensor sequence is different when we use l as angular momentum viz.
3TOO + TlO + TOl + 3Tll + 3T°2 + T 12. Actually, the spin operators commute with both the orbital angular momentum operators and the pairing quasispin operators. We have for the spin operators: S+ = N 1 3 - N 4 2 + N 5 6 ,
S_ = N a l - N E 4 W N65,
(35)
Sz = ½(N11 -- N22 - N33 + N44 + N55 " N 6 6 ) , b
and for the orbital angular momentum operators t+ = ~ f i ( N ~ + N~4-- N~2 ~36), l_ = x / 2 ( N 5 1 + N , , s - N 2 6 - N 6 3 ) , -
l~ = N l l - N 2 a + N a 3 - N 4 , ~ .
'
(36)
SCHEMATIC INTERACTIONS
137
Tensors with respect to the subgroup RL(3)x Rl(3)x Rs(3) may be denoted T~k~, where (2/0 expresses tensor properties with respect to RL(3), (k~) with respect to R,(3) and (trh) with respect to R~(3). The tensor sequence of R(12) is then R(12) = {T TM + T ° I ° + T * ° ° + T * n
+ T °zl +T*2°}.
The tensor T 12° contains E2 type quadrupole particle-hole and pairing operators, T 1°° is the pairing quasispin, T °*° is orbital angular momentum and T TM is spin. The tensor T TM contains M2 type operators and T 1.1 E1 and M1 operators. We are now mostly interested in T 12°, T 1°° and T °*°. These operators form a subgroup of R(12) which can be seen easily from commutation relations obtained by generalising eq. (4) to include spin. This subgroup is a symplectic group Sp(6) as has been shown by Pomorski and Szpikowski 2s). We denote it SppQ(6) since it contains pairing and quadrupole operators. These operators of SpeQ(6) commute with the spin operators, which follows from the fact that all T zk'r tensors of SppQ(6) have a = 0. In the same way we may form an Sp(6) algebra out of T °21, T °1 o and T TM which commute with paMng quasispin, and thus equals the Flowers Spe(6), cf. sect. 2. Taking only # = 0 components (i.e. only single-particle operators N~p) of the tensors T 12°, T 1°°, T °I° reduces Sppo(6 ) to SUpo(3), the wellknown SU(3) group of Elliott 24). Still, the group Sp(6) is cumbersome to work with and an additional simplification will be introduced. We may consider only operators of a two-dimensional N = 1 oscillator shell, in which case the threedevel model is again reduced to only two levels; we retain the levels of m t = ± 1, m s = ±½ and just the operators that connect these levels. Then R(12) is reduced to R(8). The group of pairing plus quadrupole operators SppQ(6) is reduced to SppQ(4)--~ RpQ(5), containing the T ~°° tensor, the q = 0 component of T °1° (viz. the operator lo) and the q = ___2 components of T 12°. This R(5) group has previously been explored in some detail, cf. refs. 26-28). The Elliott SU(3) reduces in the same way to an SU(2) or R(3), previously discussed by Lipkin 29). The group R(5) is sufficiently simple as to allow for a generalisafion into RpQ(5) x RpQ(5), corresponding to two sets of shells with a single-particle splitting. This problem has also been considered previously 3o) and we refer to refs. 28,ao) for the method of solution. Our Hamiltonian is now H = ~(N(1) - N(2)) - Go(L +(1) + L + (2))(/,_ (1) + L_ (2)) -- ~Gz(T,t22(1) + T~22(2))(T,*22(1)+ T,az2(2))t -kG2(T)-22(1) + 7",1-22(2))(7"11_22(1) + T?_22(2)) t
- l z ( T g 0 ) + T g ( 2 ) ) ( r g 0 ) + TO#(2))' --~Z(Tol-Z2(1) + Td2.-z(2))(To*Z_2(1) + T~_22(2))t,
(37)
138
J. K R U M L I N D E
where arguments (1) and (2) refer to upper and lower sets of shells, respectively. In the notation of ref. no) (cf. also ref. 2s)) we have
n = KNO)- N(2))- Oo v,-
zvQ
-~G2{(Vt++ Uz+)(Ua- + Uz-)+(V, ++ Vz+)(V1- + V2-)}. (38) 4,2. N U M E R I C A L
RESULTS
With the model described above we may now test the collectivity of states with respect to the various degrees of freedom corresponding to the different interactions and also the competition with the single-particle effects. The magnitudes of the coupling constants are not directly comparable with those in sect. 3 since the full rotationally invariant quadrupole type interactions were not used here. Fig. 4 shows the energy difference between the first excited 0 + state and the ground state (upper half) and the excitation energy of the lowest 2 + state (lower half) as a function of the single-partMe energy gap ~. The leftmost part of the figure shows the behaviour with only monopole pairing present (Go = 0.1, G2 = Z = 0). The state shows no decidedly collective properties at pair degeneracy D = 1 but the collectivity increases with D, i.e. the excitation energy becomes small within a certain interval o f the parameter ~. For D = 3 the state has a pairing vibrational character around
D=I
.Q=I .
.Q=I
i
to T D=3
,
,
,
t
I
0
>
.5
.5
i
ID=I /~Q=2 1
1
E2T
E2T
~ , , ' ~ S 2
i
i
i
I .5
•
0
~
~
~
=3
I
i
I
l
•
.5
[
0 I
0
.5
e
.Q=I 1
~2T~
.5
0
]
0
E2;
3
~
~
f
~
I
.5
~
i
0
0
,
:
I
I .5
Fig. 4. T h e excitation energy of the first excited 0 + state (upper part) and the lowest 2 + state (lower part) in the model described in subsect. 4.1. T h e p a r a m e t e r e is varied. The leftmost part o f the figure shows the results for Go = 0.1, Gz = Z = 0, the left central part Go = 0.1, G2 = 0.4, Z = 0, the right central part Go = 0.1, G2 = 0, Z = 0.1, a n d the rightmost part Go = O.1, G2 = 0.4, Z = 0.1.
SCHEMATIC INTERACTIONS
139
8 = 0.4. This phenomenon was previously investigated by Hoegaasen-Feldman 1) within (for this special case) essentially the same model and with the same qualitative result. The 2 + state, shown in the lower leftmost of the figure, shows no collectivity, since this state is not specifically influenced by the monopole pairing. The specific parameter determining the energy of this state is Go/Gerit where aerit ---- /~/~'~, thUS giving an increase of excitation energy with increasing e, with smaller slope if f2 is larger. The left central part of the figure shows the behaviour of these states with both monopole pairing (Go = 0.1) and quadrupole pairing (G 2 = 0.4) present but no quadrupole particle-hole interaction (X = 0). The 0 + state displays the same behaviour as with only monopole pairing present, although here the collectivity is essentially due to the quadrupole pairing interaction. The 2 + state (left central, lower) shows a less transparent pattern: for f2 = 1 there is no collectivity, for f2 = 2 this state manifests itself as a well-developed quadrupole-pairing vibration, for f2 = 3 the ground state is quadrupole superfluid and the nucleus is again less soft against changes in the coupling constants. The right central part shows the states with monopole pairing (Go = 0.1) I and quadrupole particle-hole interaction (Z = 0.1) but no quadrupole pairing. The ground state is now deformed. The first excited 0 + state (upper part) shows some collectivity at f2 = 1 (beta vibration) b u t a t f2 = 2 and 3 the interference of the monopole pairing degree of freedom washes out the structure and no information about the character of the states can be inferred from just looking at the energy. The energy of the 2 + state (lower part) increases with z, reflecting the fact that the moment of inertia decreases with increasing energy gap. The rightmost part of fig. 4 shows the energy of the first excited 0 + state and the lowest 2 + state with all three interactions present (Go = 0.1, G2 = 0.4, Z = 0.1). The interference of the various collective modes makes it diificult to draw any conclusions from looking at only energies. The 2 + states show some collectivity of quadrupole pairing and quadrupole particle-hole character for f2 > 2. The destructive interference between the pairing and particle-hole components of the quadrupole interaction makes the ground state less deformed and more soft against changes in the coupling constants than in the case of only monopole pairing and quadrupole particle-hole interactions. Thus the 2 + state shown in the lower rightmost part of fig. 4 is not of rotational character. Fig. 5 tests the sensitivity of the ground state and the first excited 0 + state to changes in the coupling constants, looking at one at a time. The upper left part shows the variation of the excitation energy with the strength of the quadrupole pairing interaction, for fixed values of Go = 0.1, Z = 0.1 and f2 = 2. Two different values of e are employed: e = 0 and ~ = 0.2. The curves show very clearly that there is a region of (72 for which the state is a low-lying collective vibration; this value of Gz is of course shifted to a larger value when ~ is larger. The lower left part shows the variation of the expectation value of the quadrupole pairing interaction itself, thus
140
J. K R U M L 1 N D E
+-
¥
tu
~=0.2
~=0.2
i .2
a
i
i
J+
.6
.8
0 0
i
i
s
1
.05
.I X
.15
.2
Gz
8
16
i ',,t-',/ ~,
~
,-
Q
.2
.0
.8
J
e=t~2
2O
0
.OS
.1 X
/ ~ ' = 0
,-, /
i t' "J \i t/ ( '7., ,t / // ~/ i t , ) .>({__.__/'..=:o5
....
.t, Gz
0 G
/ f " "
/--\
•
.15
%
<:o.2
.2
G
.1
Fig. 5. Excitation energy of the first excited 0 + state (tipper part) in the model described in subsect. 4.1. Here g2 = 2 and curves for a few different values o f e are shown. The leftmost part shows results for fixed values of Go = 0.1, Z = 0.1, while G2 is varied; the central part Go = 0.1, Gz = 0.2, while Z is varied and the rightmost part Gz = 0,2, Z = 0.05, while Go is varied. The lower part o f the figure gives the expectation values of the relevant interaction, thus showing the real correlation effects. In the leftmost part the quadrupole pairing expectation values are shown, in the central part quadrupole particle-hole, and in the rightmost part monopole pairing expectation values. Solid lines show the behaviour of these quantities for the ground state, dashed lines for the first excited 0 + state.
1.0
t.O
.8
.8
~.~ t~ J .2
0
0
.I
.2
~3
02
.4
.5
.I
.2
.3
1
I
,¢
.5
I
G2
Fig. 6. Ratio of the (t, p) and (p, t) cross section from the ground state in the n-particle system to the first excited 0 + and the ground state in the (n~2)-particle system. Here ,Q = 2, n = 4. Go = 0.1, e = 0.4, while G2 is varied. The left part has Z = 0.I, the right part Z = 0.2.
SCHEMATIC INTERACTIONS
141
testing specifically the collectivity for this particular mode. It is seen that the quadrupole superfluidity of the ground state (solid line) increases conspicuously just in the region where the first excited 0 + state has the collective character, indicating that these features are the results of a phase transition. The qnadrupole superfluidity of the excited state (dashed line) shows a slightly more complicated behaviour: it first increases when G2 increases but after assuming a maximum it decreases (for e = 0), thus leaving the larger part of the interaction strength to the ground state. At e = 0.2 this feature is less clearly displayed as the presence of the single.particle field to some extent inhibits and smoothes out the collective effects. The central part of the figure shows the same results for the quadrnpole particlehole interactions (here Go = 0.1, G2 = 0.2, Z is varied). The spherical-deformed phase transition is very clearly borne out also for rather large values of 8; the first excited state is first spherical, then of a quadrupole or beta vibrational character in the phase transition region and finally again less collective, because a large part of the strength has now gone into the ground state. The rightmost part of fig. 5 shows the results of variation of the monopole pairing coupling constant (here G2 = 0.2, Z = 0.05) for a nearly spherical system. The presence of a strong quadrupole pairing field interferes seriously with the monopole pairing and smoothes out the phase transition. Finally fig. 6 shows the (t, p) and (p, t) cross sections, displaying the asymmetry resulting from a strong quadrupole pairing interaction. Because of the different single-particle quadrupole moments of the mj = ___)levels and the m~ = -t-½ levels the pair addition and pair removal modes in a model without particle-hole symmetry (here f2 = 2, n = 4) will have different transition rates, since monopole pairing and quadrupole pairing may interfere destructively in the one mode and constructively in the other mode. This phenomenon is discussed more elaborately by B~s et aL 6) and the result of this solvable model seem to confirm their physical picture. In conclusion one may say that the various phase transitions (normal-superfluid, quadrupole normal-quadrupole supeffluid, spherical-quadrupole deformed) are more clean and usually also more sudden if the corresponding interaction is more pure. Then the vibrations can be extremely collective at certain points. The presence of other interactions smoothes out these features, just like the presence of a single-particle field does. Generally, if many interactions are present there is not much information to gain from just looking at the energies since the interference between the different modes of collectivity may sometimes make the energy of the collective state insensitive to changes in the coupling constants. In such cases the state may change considerably the character of the collectivity without showing this in the energy. Then it is essential to look at specific quantities related to the correlations of the interactions themselves, like two-particle transfer cross sections and multipole transition probabilities. The author is very much indebted to Ricardo A. Broglia for many inspiring and exciting discussions, which finally made this work possible.
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