ANNALS
OF PHYSICS
181, 79-119 (1988)
l//V Expansion
in the interacting
S. KUYUCAK
Boson
Model
AND I. MORRISON
School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia
Received July 1, 1987; revised September 18, 1987
We show that angular momentum projection in the intrinsic state formalism leads to a l/N expansion of physical quantities in the interacting boson model. The expansion parameter corresponds to the boson number which typically has values N z 10. The method is demonstrated in solving a general quadrupole hamiltonian with arbitrary kinds of bosons. Analytic expressions are derived for energies and electromagnetic transition rates of the ground-, fi-, and y-bands. Effects of the g-boson in various quantities (moments of inertia, g-factors, etc.) are discussed. 0 1988 Academic Press. Inc.
1. INTRODUCTION In the past decade, the l/N expansion has been a popular approach in atomic and particle physics. Especially in QCD, which is notoriously difficult to solve, it has provided valuable insight. Unfortunately, in both fields N = 3, and convergence of the series is very slow. Thus, making quantitative predictions is not an easy task. In contrast, the interacting boson model (IBM) of nuclear collective states [l] has a natural expansion parameter, namely the boson number, which is typically N = 10 (even larger for the deformed nuclei which is the primary concern of this paper). Truncation of the series at the l/N* level is quite sufficient for quantitative results. Hence the l/N expansion is well suited to IBM calculations. Currently, there are two avenues available for the IBM calculations. Computer codes can be used for numerical diagonalisation of a general IBM hamiltonian [2]. Or, in special cases corresponding to dynamical symmetries of the hamiltonian, one can find analytical solutions to the model problem, and then use perturbation theory to treat deviations from that symmetry. Clearly, when available, analytical solutions are preferable to numerical ones, and even more so as the model gets more complicated. There is both microscopic [3] and phenomenological [4, 53 evidence that the simple sd-boson model is not adequate to describe deformed nuclei and must be extended to include g-bosons. Since the sdg IBM hamiltonian contains many more parameters (32 compared to 9 of the sd-model), selection and determination of a simple set of parameters through numerical analysis are difficult. Algebraic solutions of relevant model problems would certainly shed some light on this process. 79 0003-4916/88 $7.50 595:18lil-6
Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.
80
KUYUCAKANDMORRISON
The deformed nuclei correspond most closely to the SU(3) limit of the IBM and much analytical work was done based on this limit. However, careful phenomenological analyses of the deformed nuclei indicate that this limit is far from being realized [6]. Values of the parameter x in the quadrupole operator range from -0.4 to -0.6 [7] to be compared with the SU(3) value of -1.32 and the O(6) value of 0. Thus a perturbation treatment of deformed nuclei based on the SU(3) limit is not practical, and one must go beyong the group theoretical techniques for algebraic solution. In the following, we will show that angular momentum projection in the intrinsic state formalism leads to a l/N expansion of matrix elements. Since variation after projection (VAP) is equivalent to the Schrddinger equation, this provides an algebraic solution for an IBM hamiltonian. The method is easily generalised to arbitrary kinds of bosons, hence allowing study of the effects of the g-boson in realistic situations (i.e., away from the SU(3) limit). Mean field techniques for boson systems were previously introduced in the context of the classical limit of the IBM and its relationship to the Bohr-Mottelson model [lo]. Following this, intrinsic states were used in estimating electromagnetic transition rates in deformed nuclei [ 111. The next step of angular momentum projection was first tackled approximately in the self-consistent cranking model [ 121 and more recently, exactly albeit numerically in connection with fermion-boson mapping [13]. As will be shown in the text, it is possible to carry out angular momentum projection algebraically for boson mean fields with a reasonable effort and to obtain analytical expressions for the energies and electromagnetic matrix elements. In Section 2, we introduce the general formalism. Calculation of energies for the ground-, /?-, y-, and other K-bands is presented in Section 3. Algebraic expressions for intra- and interband electromagnetic transitions among these bands are derived in Section 4. Section 5 gives a summary and concluding remarks. Throughout the text, the following shorthand notation is used for angular momentum eigenvalues: L = [ 2L + 1 ] ‘I2 and L = L(L + 1). The M-quantum number is suppressed as an argument or index when M = 0.
2. FORMALISM
We will demonstrate
the method in solving the hamiltonian H= -KQ.Q-dL.L,
(2.1)
which has been successfully applied to the deformed region in the &IBM numerically [6, 71. Since the L . L term is always diagonal, it has no effect on the wavefunctions and will be dropped henceforth. It can be easily restored by adding - dL( L + 1) to the final energy expressions. We introduce the boson creation (6k) and annihilation (6,,,,) operators, where
l/N
I = 0, 2, 4, ...) p correspond
appropriate
EXPANSION
81
IN IBM
to s, d, g, ,..,-bosons, and m is the projection boson operators are given by
axis. Intrinsic
b~=~x,b~m, I
~(xlm)*=x, I
‘X,
= 1,
on an
(2.2)
where m =O, 1, 2, .... p. The quadrupole operator in (2.1) can be generalized to arbitrary kinds of bosons by defining a parameter matrix q of order 1 + p/2,
0 402
0
.
.
o
q42
q44
q46
’
.
.
.
.
.
(2.3)
The matrix q is symmetric which is written as
to ensure the hermiticity
of the quadrupole
operator
(2.4)
In the usual notation, qo2= 1, q22= x in the sd-case, and qo2= 1, qz2 = /3, q24= y, qa = 6 in the sdg-case. For a general intrinsic state 14) = (qp
(by’.
the structure coefficients x,, are determined tion (VBP),
or after projection
. . (by
I-),
by varying (H)
(2.5)
either before projec-
(VAP),
(WI. = (41 HP;, l4>/($l&&
Id>,
(2.7)
where D&(Q)
R(Q) dfi2
(2.8)
is the projection operator. Previously, we have shown that for the low-lying states, which are dominated by the ground-band intrinsic operator (i.e., No B N,, m # 0), both methods give the same result for the variation to leading order [S]
&- CA(O)A(m) + (1- dmo)B*(m)1 = Lx,, Im
(2.9)
82
KUYUCAK
AND
MORRISON
where
A(m)=x(-Y” 9
Ciml-m
120)qjrXj,,,Xt,,,,
(2.10)
B(m) = c (jm 10 Pm> q,rximxm. iI In (2.9), the second term represents quadrupole mixing of b, in b, and 6,. B(m) vanishes in the W(3) limit (see Appendix C) and is small in general. Neglecting it, Eq. (2.9) becomes equivalent to the eigenmode condition [S] (2.1 la) giving 1
t-1”
(jm
1-m
120)
qj/Xjnz
(2.11b)
=&X,mr
which provides a convenient zeroth order solution to all the intrinsic boson operators. Equation (2.9) is an eigenvalue equation, and the number of solutions depends on the order. For example, in the sdg-IBM, Eq. (2.9) has order 3 for m = 0, order 2 for m = 1,2, and order 1 for m = 3,4. Thus there are three solutions for m = 0 denoted by b$, bht, bit (with eigenvalues 1, > &, > 2:) which correspond to the ground-, fi-, and b’-band intrinsic operators. As is shown in the next section, Eq. (2.7) results in l/N expansion of (H),. For higher order corrections in the intrinsic boson operators, one should evaluate (H )L with appropriate intrinsic states to the desired level of accuracy and then apply the variational principle. Once the intrinsic states are determined matrix elements of other operators can be calculated using (Appendix D) (K’L’JJ T”’ ML)
=2[q4K,
L’(2L+ 1) L’) J/-(lbK, L)]‘/’
c (LMIiq MN
L’R)
where E = [2L + 1 ] ‘I*, #K is the intrinsic state for the K-band, and Jlr(d,, denotes the normalization of the projected intrinsic state, given by
L)
(2.13) The leading order equivalence of intrinsic state matrix elements with the projected ones (e.g., Eqs. (2.6) and (2.7)) is in general true for all intraband matrix elements. This equivalence is also carried into interband matrix elements involving the ground-band as is shown in Section 4.
1jNEXPANSION
3.
a3
IN IBM
ENERGIES
3.1. Ground-Band
The ground-band
intrinsic state is given by
1
I&>=[ (b$)N+ (b&)N-2 c LJ$bt,f ... I->. m+O
(3.1)
The first term in (3.1) already includes mixing of all the K = 0 bands. The second (and higher) terms represent mixing from K# 0 bands. Due to the dominance of the ground-band operator and cancellation with the normalization, such terms usually contribute at l/N* or higher level. Henceforth, we will neglect the higher terms in the intrinsic states unless they make a contribution at the level considered. Also, for convenience, we will suppress the subscript m if m =O. With these provisions, Eq. (2.7) gives for the ground-band 2L+l (Q.Q)L=2N,4p,
L)
2Lf1 = 2Jwg,
b”Q . Q%o(b+)” I->,
(3.2)
dP sin P G,(P) L) s
(-I b”Q . Qe-iSJY(bt)N
x
where the normalization
(-I
I-),
(3.3)
from (2.13) is 2L+l
M(qSg, L) =2
1 d/? sin fl d&,(p) (-I bNe-iBJy(bt)N
I-).
(3.4)
Defining the rotated instrinsic operator as (3.5) one can calculate the matrix elements in (3.3) and (3.4) as <-I b Ne-iBJy(bt)N
I-)
= N!
=N! [Tx;d&(/?)]*.N!
[Z(p),“,
(3.6)
84
KUYUCAK
AND
MORRISON
(-I bNQ.QciPJv(b+)" I-)
+=w)cjl & (4jIxl)2 dkil]y
(3.7) where in the last step, we have used angular momentum algebra to combine the d-matrices and to sum over the tree C- G coefficients. Exact evaluation of the ,&integrals is difficult and also forces one to specialize to a specific boson model (i.e., sd, sdg, etc.). Instead, we employ a gaussian approximation to [Z(p)]” which is valid for large N, and has the further advantage that the expressions will remain general, [Z(P)]”
N (x .x)” e-y
r=l
with
YN
1 JJ= z c i$/x I
. X,
(3.8)
where i= 1(1+ 1). The exponential in (3.8) is negligible for /I > 7c/2. Extending the integration limit from 7c/2 to co, the B-integrals have the general form (argument M is suppressed if M = 0) F(T’, L) = loa dfl sin /I d&,(B) e-B2’r, which is evaluated in Appendix A. Substituting the normalization
(3.6) and (3.9) in (3.4) we obtain for
A’-(~S,,L)=(~L+~)N!(X.X)~F(I-,L). Combining
,,
(3.9)
(3.10)
the d-matrices in (3.3) and (3.7) and using (3.9) give the following for
l/h’
EXPANSION
85
IN IBM
(Q.Q)~=~~~)sNN!(x.x)“l t?’ X
EC
(pj’o
1 JO)(lOl’O
] JO)
W /I
x qj,qi~~xjxj’x,x~
c (JO LO [ ro)2 F(I-2, I)
* (q,rx,)2 +ci, 2it 1
where r, =2/-v(N-n).
Substituting
j' j J i I1’2
(3.11)
c (10 LO ( lo>2 F(T,, I) , I 1
(AS) and (3.10) in (3.11) leads to c (flj’0
I JO)(fO
I’0 ) JO)
/I’
x~[l-~~S,(J,L~+~j+~(S,(J,L)+ZS,(J,L~+~~
+-&
.
i
S,(.J, L)+4S,(.J,
L)+ +l(J,
S,(J, L)+$,(J,
L)+
‘-5
L)+$
1 1 + .. .
)
(3.12)
where the sums S,(L,L’)=~(P)“(LOL’O~10)2, I
(3.13)
are evaluated in Appendix B. At this point, it will be useful to discuss the general form of the l/N expansion for the expectation value of a k-body operator 0, (3.14)
86
KUYUCAK
AND
MORRISON
Clearly, the leading term C, is independent of projection. For the terms with n + m = 1 (i.e., C,, and COi), only the first terms from powers of l/N are needed in F(f, L) (AS). For the second layer of terms (n +m =2), the first two terms from powers of l/N are needed, and so on. Here, we terminate the expansion at the second layer. The sums over J in (3.12) can be done using (B.5). Noting that the J independent terms just give back F(T,, L) and regrouping the terms proportional to 1 and J2, we get from (3.12)
x
f+4iT’+P’-;(l+?-I)(j-l-6)(j’-P-6) (
+-&i2_&12(2[+3)-2(~+5)(i-~-6))(~’2-i’2-12(2?+3) -2(f’+5)(j’-P-6))
(3.15) In order to simplify (3.15), we introduce, A,,,,, = 1 jmi%jj,xjxl,
Sjl
= GO 10 120)
qjlv
jf
(3.16)
c, = c -AP(qj,x,)2, j[ 21+ 1
in terms of which the coefficients of the second and third terms in (3.15) can be written as D, = (2A, - 3A)A, Dz =h(Az
--A,,)‘-(A,
-A,,)A,
+2A2A+7A;-24A,A+18A2.
(3.17)
l/NEXPANSION
87
IN IBM
Note that the subscripts m and n are supressed whenever they are zero. In the last stage, we express Eq. (3.15) in the form of (3.14) using the expansions
(3.18)
The final result is (x .x suppressed for convenience) AZ+1
(Q.Q),=Nz
{ D,+C-c, + 8y2
In the W(3)
limit
A2-D,+C
N[
2~ +L 2~ yN2
2
+’
1
N2[
1 [
-A2+
A22
1-L
(
31.1
l-
29(
1
dD1 (3.19)
$+$(-?(+)A2
of the sd-boson model, Eq. (3.19) gives (see Appendix C) (Q.Q),=ZN2+3N-f-t
;+kN (
+-&$. )
(3.20)
The C,, C,,, and Cl0 terms correctly reproduce the SU(3) energy formula but the Co2, Cl1 , and C20 terms fail to vanish. This is related to the gaussian approximation which fails in the second and higher terms of each power of l/N (see Appendix A). In the vicinity of the W(3) limit, it is better to use the integral formula obtained in that limit. Employing Eq. (A.6) instead of (A.5) in the foregoing calculation one obtains
88
Equation vanish in other and Finally, expression
KUYUCAK
AND
MORRISON
(3.21) differs from (3.19) only in the Co2, C,,, and C,, terms which the SU(3) limit. The two equations, (3.19) and (3.21), complement each either one can be used depending on the region considered. in the special case of the sd-boson model, it is possible to obtain an exact using the exact integral formula (A.9),
(3.22)
Comparing the terms Co2, C,,, and C,, in (3.19) and (3.22), it will seem that the gaussian approximation is doing much better than one would expect (cf. F(f, L) in (A.5) and (A.9)). In fact, F(& L) enters both the numerator and the denominator which mostly cancel out for the approximate second terms. 3.2. Variation qfter Projection In applying the variational principle, we will limit ourselves to the terms C,, C,, , and C,, for simplicity. Since these terms are independent of the approximation used, the results will be exact to that order. Restoring the factors of x . x in (3.19) and varying with respect to x, gives the set of equations for I = 0,2, .... p,
(3.23)
Equation (3.23) is amenable to order by order solution. The leading order was discussed in Section 2 and gives the eigenvalue equation (2.1 lb). Denoting the solutions of (2.11b) by xp, for higher order terms we try
x,=x:+$+& Substituting
I = 0, 2, ...) p.
(3.24)
(3.24) in (3.23) leads to the following equations for yI and z,, 2 AF-2A2x [ 8x1
=---
1x0+ YIN
I
2AA,-3A2+ 2y(x.
x)’
C r-i
1 #’
(3.25a)
EXPANSION
l/N
2A2x
Evaluating
’
89
IN IBM
1
x”+zt/NZ
the derivatives, we obtain from (3.25)
(i-317-
x
A2 2Aa, - 312 i-
[
2Y2
y&Q:++:. I
(3.26a)
and p = C(x”). The set of linear equations (3.26) where I = A(x’), rs,, = A,,(x’), and (2.11b) completely determine the structure coefficients (x7, y,, z,} in terms of the quadrupole parameters qi,. Although for a specific boson model these equations can be solved analytically, they are more suitable for numerical solution. A number of interesting observations can be made, however, without actually solving (3.26). In the SU(3) limit, the r.h.s. of both (3.26a) and (3.26b) vanish whereas the determinants on the 1.h.s. do not. Hence the structure coefficients (v,, zI> are all zero in this limit (and only in this limit), which shows that the SU(3) limit corresponds to the absolute minimum of the quadrupole hamiltonian (2.1). In the sd-boson model, the same thing happens to Eq. (3.26b) in general, and the coefficients {z,} vanish identically for all values of q,,. Thus, the structure coefficients (x,} are independent of L in the sd-boson model using a simple Q . Q interaction (2.1). In more intuitive terms, the sd-boson system does not respond to the rotation by changing the character of the intrinsic state. Apart from the special cases mentioned above, the coefficients {xl} depend on L, and the boson system, in general, exhibits stretching. The final energy expression for the ground-band is obtained by substituting (3.24) in (3.19) limited to the Coo, COl, and C,, terms. Since x”~xo+~xo~y+~txo~z)=ixx,
(3.27
to that order, the energy formula remains the same, 220, - 3A2 2Y
+p
220, -3a2 4Y2
> 1 + ...
.
(3.28
90
KUYUCAK
AND
MORRISON
The effect of the variation shows up only in second and higher layer terms (i.e., Co*, Cl,, Cz,,, etc.). Clearly, this is true for any quantity whose leading terms is A (or A2), which includes energies, quadrupole moments, and intraband E2 transitions. 3.3. /l-Band
The j-band
intrinsic state is given by I&> = [(b+)“-’
b’+ + (b+)N-2
c
t;b;bt,
+ ...I
I-).
(3.29)
rn#O With the expansion terminated at the first layer, only the first term in (3.29) is needed. Substituting in (2.7) gives
(Q.Qh
=
Bd&b(B) J4sin 1b’tI-), x(-Ib 2L+l
2@4%$JPL)
N-1brQ.
Qe-iB”Y(b+)“-
(3.30)
with the normalization
Jdj? sin /I d,L,(j?)
J(T(qbB, L) = y
x (-I bN-lb’e-ifl-‘y(b+)N--
b’+ I-).
(3.31)
Defining the rotated intrinsic states b L, b’k as in (3.5), we calculate the matrix elements in (3.30) and (3.31) following steps similar to (3.6) and (3.7), (-1
bN-
lbfe-iBJy(bt)N-
= (N-
1 b’+ I-)
1
l)!
=(N-l)!ZN-‘[Z’+(N-I);], (-1 &‘“-
‘b’Q
= (N-
. Qe-“-‘y(b+)N-
l)(N-
1 b’+ I-)
l)! ZNp3(-1
(3.32)
l/iv EXPANSION
+z
IN
91
IBM
[(
z +z--+Y& 1&I a
Q.Ql->
N-lab;
=5(N-
l)(N-
+ 4 yxjxj.x;xl.
l)! ZN-3
22 +N-2
+zcl
jr -21+ 1 q; dbo
i
(N-2)
1 (joj’0
IJO)(lO
I’0 IJO)
Jjj
II
x;xj(xlx;,
+ xix/,)
[(
Z’+(N-2);
1
x:+2yx,x;+-x
where xi denote the structure-coefficients Z(x’) (see (3.6)) and Y is
Z N-1
for the P-band intrinsic
”
11 ’
(3.33)
operator, Z’= (3.34)
Y(B) = 1 XIX; 4&V
To evaluate the /I-integrals, we use the gaussian approximation (3.8) for the powers of Z(b), and combine all the d-matrices (including the ones in Y and Z’). The normalization (3.3 1 t( 3.32) becomes J’Q,,Z,)=(2L+l)(N-l)!(x.x)-’
N-l x Cx;‘C (lOLO po)*F(z-*, z)+--~x,x;x~x;~ [ I I x~(101'O~JO)2(JOLO~ZO)2F(Z-z,Z) IJ
Substituting (A.5) for F(f, I,) and evaluating get from (3.35) X(4+)=(2L+l)(N-l)!
1 .
the sums with the help of (B.4), we
(x-x)“-lx’.x’
x
+(N-1):
(.
$4-s
.
36z+
-.)x.xT,.x,],
(3.36)
92
KUYUCAK
AND
MORRISON
where we define (some for future reference) a’ = c ix;‘, /
a,,, = c lx;,,,, I
b, = c 1”2x,x,, , 6; = 1 P’*x,x,~, I I
b = c h,x;, I b2 = 1 [l(rI
(3.37) 2)]“*
xIx12.
Note that because x . x’ = 0, each factor of Y effectively brings in a factor of l/N. Thus, the last three terms in (3.33) do not contribute at all at the level considered. Substituting (3.33) in (3.30), we get (@Qhu
2L+l =M(4p, L) 5(N-
l)(N-
l)! (x.x)--2
N-2 Gz
X
(JOY0 IJo> {: II x xlxyxp 1 (x,x;* + 4xix,xh) 1 k
x c (M) k’0 IJlo)* (JO.z,o1J20)2
:f ;}
C?jSZfl
(J*O LO iZO>* F(;(r,, I)
JtJ2f
+A
x;x,,(x,x;,
+c-jr 21+’ 1
+ x;xr)
k
x (.ZIO LO lZO)* F(T,,
Evaluating the sums involving J using (BS), we obtain (Q.Q),,,
=N(48,
2L+1
X
Ec
L)
c (JO LO IZO)’ F(T,, I) I
1
JII
(3.38)
I) .
the C - G coefficients using (B.4) and the sums over
(N-l)(N-l)!(x.x)-’ (fl lo 120)
qj/fij'l
l/N EXPANSION
i~-&qj,x,)‘~(l-~~+
+x’.x’C
93
Iti IBM
.-)].
(3.39)
By virtue of Eq. (2.1 lb) and x . x’ = 0, the second term and part of the fourth term vanish. Substituting the simplifying expressions (3.16) and dividing by the normalization (3.36), we obtain from (3.39) =N2
(Q.Q)a,L
A’f;
-A2+2AA’-2AA;;3A2+C]
i
[
+LJ 2!?+2**;;3*‘]+ ...I. Y
(3.40)
where A’ = A(x’). Equation (3.40) is exact (at the level considered) and correctly reproduces the W(3) energy formula. As discussed in the last subsection, variation does not affect the results, to this order, so they will not be considered. However, noting the equivalence of the Coo and C,, terms in (3.19) and (3.40) variation of (3.40) would still give the same coefficients {z~} as the ground-band (3.26b), but would give different results for { yl}. The final energy formula for the P-band is then
+E
--A2 Y 2&J,4Y2 3A2 + ..-
(
> 1
(3.41)
Calculation for the /Y-band follows the same steps as above with the replacement x’ + x”; hence the energy formula is obtained from (3.41) by substituting 1” for 1’.
94
KUYUCAK
AND
MORRISON
3.4. y- and Other K # O-band
The intrinsic state is given by 1~~5~)= [(by)“-’
bt,+ (b+)N-2
c 5fbLbk-,+(K-t rn#O,K
-K)+
...
1
I-).
(3.42)
Again, terminating the series at the first layer, only the first term in (3.42) contributes (b-, gives the same result as b,, so it is ignored). Substituting in (2.7) gives
PGAP) J4sin bf:I-), (3.43) x(-Ib Qe-i@y(bf)N-1 2L+l
<~.Q>,.L=~~(~~,~)
N-lbKQ.
with the normalization ~(4,,
Jd/? sin /3 d&(/I)
L)=?
(--I bN-lbKe-iBJy(bt)N--l
bS, I-).
(3.44)
Introducing the rotated intrinsic states b $, btR as in (3.5), calculation of the matrix elements in (3.43) and (3.44) is similar to that in the Subsection 3.3 (i.e., b’ --t bK), (-I bN- lbKe-ifib(bt)N--l
= (N-
bj, I-)
i)! ZN-’ i
(-1 bN-
l&Q.
z, + (-)” (N-
1) 41,
(3.45)
bf: I-)
Qe--ifi+(b
=5(N-l)(N-l)!ZN-‘{(N-Z);{;
;, ;lqj,qY, II’
Z,+(-)K(N-3)2 x (joj’0
IJO)(lO
+ 4(-)K YK($ +r&
I’0 IJO) xjx,-xlxr
d&
j’0 [JO) (1K 1’0 JJK) XjXi,X/KXr dh
(jK j’0 (JK) XjKXj,
x ( (10 I’K (JK)
xIxrK + (1K I’0 (JK) xIKxP) d&
x2, d’@-J
[( + 2(-)K Y,x,x[,
1
d’Ko + -
N-l
&d!w
I> 7
(3.46)
l/N
EXPANSION
95
IN IBM
where 2, and Y, are
Z,(P)=c x&d$yK(B), YK(b)
=
c
xlxlK
(3.47) (3.48)
dkO(h.
Next, we evaluate the p-integrals using the gaussian approximation powers of Z(b), and combining all the d-matrices. The normalization becomes Jlr(~K,L)=(2L+1)(N-1)!(x.x)N~’ +-
N-l
Cx:,C(1KL-K110)*F(T,,z) [ / I
c,, x,xIKxrxrK~
x.x
(3.8) for the (3.44)-(3.45)
(10 1’0 IJO) IJ
1
x(lKl’-K(JO)(JOLKIIK)2F(T2,1,
K) .
(3.49)
Substituting (A.5) for F and evaluating the sums with the help of Appendix B, we obtain from (3.49)
K*
+
N-l X.XXK
-2
9L2+6
.(
r:
+b$
2
4z-2K2+4;
‘XK
---9t) 2.4*
1
(
b’
-2K*-1+6;
1-; 3!42
I
_ L
) )I
6,,, + . ..I}.
a,,
(3.50)
where aK and b, are defined in (3.37). Observe that PK brings in a factor of l/NK; hence the second term in (3.49) contributes only for K= 1,2. Similarly, the last two terms in (3.46) do not contribute and will be ignored. Substituting (3.46) in (3.43), we obtain S(N-
xi=;
{f II
l)(N-
l)! (x.x)“-*
{T i} qilqfl
96
KUYUCAK
x
AND
MORRISON
I'0 IJO) xjXfXIxI
[ x Cx&C ( K
+-
N-3 x-x
(kKJ0
IJ,K)Z
(J*KL-K)z0)2F(f,,
I)
IJl
ckk’ xk~kK~kr~k.K
x (k0 k’0 \J,O)(JO +d(joj’o
c
(kKk’-
J,O \J,O)2
IJo)(lKro
K IJ,O)
JlJzI
IJK)
(J,OLK
\ZK>2 F(T,,
Z, K)
1 xkxkK
dyjxfx/KXr
k
+& + (Xl’0
(jK j’0 1JK) xjKxf( (IO I’K 1JK) (JK)
x,,.q)~
(JKL-KIZ0)2
xl?TrK
1
F(f,,l)
I
+cL(q,,*,)z[~4,; jI 2f+ 1
x (J, K L - K
+-
N-2 x.x
c kk’
I
IA)' F(f,,
XkxkKXk’xk’K
I) c
(kK k’ - K )J, 0)
J,J>I
x (k0 k’0 (JIO)(ZO
Evaluating the sums involving over J using (B.5) gives
J,O (J,O)*
(J,O LK lZK)2 F(T,, Z, K)
Ii
.
(3.51)
the C - G coefficients using (B.2)-(B.4) and the sums
2L+ lL) (N- l)(NK.L= N(4K,
l)! (x.x)“-’
(fl 10 (2O)(j’O1’0
120)
XjXfX/Xr
II F(T,, L)-K2&
-2+&3L
i+I'-$(j-i-6)(j'-F-6)
r?
-
I>
l/N EXPANSION
+(jKlO
IN IBM
(l-SL 4b.l+
~2K)(j’Ol’K(2K)x,xr,)x,,x,~
.
(3.52) Before proceeding further, we address the question of spuriousness regarding the There are p/2 solutions for the K= l-band intrinsic operator. In the W(3) limit, one is associated with the representation (pN - 2, 1) which is spurious and the rest correspond to the physical representations (pN - 6, 3), (pN - 10, 5) etc. Away from the SU(3) limit, these bands are mixed and one must project out the spurious components. This lengthy and complicated procedure can be circumvented, however, by noting that the normalization for the spurious state should vanish. From (3.50), for K= 1 this amounts to K= l-band.
112
l-h:=o+~P~zx,x,, 2Y I which, by inspection, has a solution
= crx: ) [ I 3
[ [1
(3.53)
112
X/l,
595/181/l-7’
=
-
2Y
X/3
(3.54)
98
KUYUCAK
AND
MORRISON
where the subscript s denotes the spurious solution. solutions {x,r) be orthogonal to {xII,) leads to
b,(x,)=c P2X$,1= (2y)“2 I
Demanding
c X,lsX,, = 0. I
that the other
(3.55)
Thus, with the elimination of the spurious band, all the complicating terms proportional to b, disappear from the expressions (3.50) and (3.52). Substituting (2.10), (3.16) and dividing by the normalization (3.50), we obtain from (3.52) (Q.Q>K,L
=N2
i
A’f;
[
-A2+2AA(K)+2B2(K)-2AR;;3R’+C] (3.56)
Equation (3.56) is similar to the corresponding expression for the b-band (3.40) (B(K= 0) = 0 because of x . x’ = 0), and all the observations made there also hold for the K-bands. The final energy formula is given by -A2+2&+2+ +L
A2 2As-3A2 -(Y 4y2
2Aa - 3A2 2y +p
1+ ..’ 19
(3.57)
where 1, = A(K, x”,) and rcK = B(K, x0, x”,). The same formula also applies to other single-phonon excited K-bands (e.g., y’-band) with appropriate changes in A, and rrK. 3.5. Discussion of Energy Systematics The general features emerging from the l/N expansion (3.14), for the bands dominated by the g-band intrinsic operator, can be summarized as follows. The leading term for each power L” (i.e., C,,) is the same for all bands. The following terms Cnl, Cn2, etc., which come at orders l/N, 1/N2, .... are band specific and distinguish between different bands. Thus, the band excitation energies are given by the Co, terms with l/N corrections coming from the Co2 terms. From the previous subsections, we have to leading order E,(K)
= NCW
- 2,) + 2713,
(3.58)
for the energy of a single-phonon excited K-band. Here the subscript v distinguishes between bands having the same K-quantum number. In Figs. lL4, Eq. (3.58) is plotted against the quadrupole parameters in the sdand sdg-boson models. In all cases, qoz is set to 1 and the remaining parameters are varied between 0 and the SU(3) value which is normalized to 1. Figure 1 shows the
99
l/N EXPANSION IN IBM
so-
30-
0
0
’ 02
’ 04
06
’
5
08
’
b
10
922
FIG. 1. Excitation energies of the fi- and y-bands in the sd-boson model as a function of qz2 (normalized to 1 in the SU(3) limit). The numbers on the curves denote the K-quantum number.
variation of the /I- and y-band energies in the s&model with respect to qz2 which is related to the deformation. Phenomenological analyses indicate that qz2 has values 3&45% of the W(3) value [7]. In the sdg-model, there are three independent parameters, qz2, qz4, and qd4. Figures 24 show the variation of energies for the lowest K-bands with respect to each of these parameters while the other two are kept constant at the W(3) values. For different choices of constants, the scale and the amount of splitting between bands change but the general behaviour remains the same. From Figs. 2 and 3 it is seen that q22 and qz4 mainly provide splitting between the /3- and y-bands and the K= 1 and 3-bands, while the relative splitting
BO-
-2 \ w*
0
0
02
’
*
04
’
06
’
08
’
10
922
FIG. 2. Excitation energies of the K= 0, 1, 2, 3, and 4-bands in the sdg-model as a function of q2>.
100
KUYUCAK AND MORRISON
FIG. 3. Same as Fig. 2 but for q14.
among the K= 0, 3, and 4-bands is mostly unaffected. In turn, qu causes large splittings among the K=O, 3, and 4-bands, and has a relatively minor effect on the K= 1 and 2-bands (Fig. 4). This behaviour can be understood in terms of the structure of the intrinsic operators; for K= 3,4 they are pure in g-boson and for K= 0 they are dominated by it; thus only qd4 is effective in splitting these bands. It is interesting that the (sdg) counterpart of the B-band in the &model is the /?-band (not shown here to avoid cluttering), and the P-band of the sdg-model has a very different structure. This may offer an alternative explanation for the weakness of the p + g E2 transitions. Some restrictions can be put on the quadrupole parameters from the energy
FIG. 4. Same as Fig. 2 but for qM.
l/NEXPANSION
101
IN IBM
systematics (E,, > EZ, E, > E,); qz4 should stay in the vicinity of the SU(3) value, whereas q22 and q44 should have much smaller values. Changing the sign of q24 does not affect the excitation energies, but does so in the case of q22 and q44. Since changing the sign of both q22 and q44 is equivalent to an overall sign change, it is sufficient to consider only one of them. In Fig. 4, for negative values of qu, the energy curves smoothly extend to the left. Clearly, the resulting band structure is in contrast with the experimental situation (E, > E,), and hence the negative range of the parameters is unlikely to be relevant. The other physically important quantity is the moment of inertia which is proportional to the inverse of C,, C,,/N”. Thus, to leading order, all the bands have the same moment of inertia given by
1
21 1 A2 2La, - 3A2 -l 7=[ -. 4y* fi K y
(3.59)
In Fig. 5, Eq. (3.59) is plotted against the quadrupole parameters q for the sd- and sdg-models. Since deformation increases with q, the moment of inertia is also expected to increase with q. The contrary behaviour of the sd-model is corrected in the sdg-model. We note that the quadrupole hamiltonian (2.1) describes a strong + weak-deformation (SU(3) + O(6)) transition of the boson system and does not necessarily require single-boson energy terms. Nevertheless, adding such terms perturbatively to the hamiltonian (2.1) would not change the above conclusion. As a final example, we discuss the CzO term which measures the deviation from the L(L + 1) rule. Experimentally the quantity - ( C2,/CI,)( 103/N2) is on the order
a0
I
,
,
,
,
,
,
,
FIG. 5. Comparison of moments of inertia in the sd- and sdg-models. The quadrupole parameters
qz2,qz4,and qu are simultaneously increased from 0 to the X43) value (normalized to unity).
102
KUYUCAKANDMORRISON
of unity whereas the sd-model gives values around 0.1 from Eq. (3.22). In the &g-model, the situation is improved; however, due to the approximate nature of C2,, in (3.19) it is difficult to give a quantitative estimate.
4. ELECTROMAGNETIC
TRANSITIONS
In this section, we calculate electromagnetic properties that have experimental relevance. These include various intraband and interband E2 transitions, quadrupole moments and g-factors for the ground-band, and Eo, Ml, h43, and E4 excitations of the corresponding K-bands. In order to limit the number of parameters, we employ the consistent Q formalism [7]; i.e., the same parameterization of the quadrupole operator (2.4) is used in both the hamiltonian and the E2 operator Tg2) = a2QM,
where a2 is an effective E2 charge. 4.1. Quadrupole Moments
Substituting the first term of the ground-band intrinsic state (3.1) in (2.12), we have for the reduced matrix element of the quadrupole operator
x (-I bNQp,e-ipJ~(bt)N
I-).
(4.1)
The matrix element in (4.1) is given by (4,I Q-,,,ediBKv 14,) = NN! ZNp’(-)“’
1 (jOI-
A4 (2 -M)
qj,xjxldLO.
(4.2)
iI
Substituting the normalization (3.10) and (4.2) in (4.1), and after performing the usual steps of (i) combine the d-matrices, (ii) sum over the C- G coefficients, and (iii) evaluate the B-integral using the gaussian approximation, we obtain
(4.3) The sums over J are evaluated in (B.5). Substituting dividing by F(T,L) give
the expressions in (3.16) and
l/N EXPANSION
(Lll Q \lL) = Ni?(LO
103
IN IBM
20 JLO)
L
A+
-2ylv2
A* -A,,
- lOA, + 12A 8~
)+.‘.1.
(4.4)
Equation (4.4) is still general and one can use a different parameterization q’ for the E2 operator by simply replacing q-i, -+ q;., in evaluating A,, (3.16). Recalling that when the leading term is A, VAP does not introduce any corrections at this order, Eq. (4.4) is exact and reproduces the SU(3) formula of the sd-boson model [9]. Assuming the consistent Q formalism, we can replace A and A,,,, in (4.4) with their respective values 1 and rrmn. Substituting (4.4) in QL=u2E;<
Q IL>
LL 2 0 ILL)
and inserting in the appropriate moments
(4.5)
C- G coefficients, we obtain for the quadrupole
n,=-a,&&["'+("-%)
o* - CT11- lOa, + 122 8~
4.2. Intraband
)+...1.
E2 Transitions
From (2.12), the E2 transition
matrix element for the ground-band
(2L+5)“*(2L+l) L+ 2) N(f&,
(L + 2” Q ‘IL) = 2[Jv(&,
L)]“Z
bNQ-Me-iPJp(bt)N
Equations (4.7) is similar to (4.). Making geometrical factors, we obtain
the appropriate
QjIL)=NL(LO2O(L+20) - L(L + 3) A-A2-A,, 2yN2
is given by
c (LM2-MIL+20) M
x s d/? sin /? dL,,(fl)(-1
(L+2(1
(4.6)
I-).
(4.7)
changes in the
A-9) +6A, 24~
-12A
>+...1.
(4.8)
104
KUYUCAK
In the consistent Q formalism,
AND
MORRISON
Eq. (4.8) gives the following B(E2) value,
As a second example, we calculate E2 transitions between the p-band members. Replacing the ground-band intrinsic state in (4.7) with that of /?-band (3.29) gives CL+211
(2L+5)“2
(2L+ 1) L)&
Q WB = 2[Jv-($h,, L+2)Jv-($p,
(LM2-M’L+20)
xs 4 sinB4h-,(B) x (-I &-’ b’Q-,e-iflJ,(bt)N--l
b’t
(4.10)
I-).
The matrix element in (4.10) is similar to (3.33) and is given by
(4pl Q-me -i~J+j&+(N-1)(N-1)!Z~-*(-)M~(JIz-M~2-M)qj[ x +
[(
Z’+(N-2);
Y(XjX;
+
Xix/)
Substituting (4.11) in (4.10) and following dividing by the normalization (3.36)
- L(L+3) 2yN2 In the consistent Q formalism,
-
A/42
iI
XjX/
Z + N-l
x,'X;
1
(4.11)
d$o(P).
the familiar
steps, we obtain
-A,,
-12A
+6A, 24~
after
) 1 + ...
. (4.12)
Eq. (4.12) gives for the B(E2) value
o2 -cl1
> ...1
+6a, - 121) + 24~
?. (4.13)
l/NEXPANSION
105
IN IBM
Comparing Eq. (4.13) with (4.5), we see that the only difference is in the second term: 1 is replaced by A.‘. It should be obvious from the last section that the B(E2) values for the other K-bands can simply be obtained from (4.13) by substituting A’ + I,. Both Eqs. (4.9) and (4.13) are exact at this order and reproduce the N(3) formulas of the sd-boson model [9]. 4.3. Interband
E2 Transitions
We now treat the E2 transitions among the ground-, j-, and y-bands, which have become a significant feature in distinguishing collective models. Cancellation with the normalization, which has prevented contribution of the higher order terms in the intrinsic states so far, does not work in the case of interband transitions. Thus, substituting the full b-band intrinsic state (3.29) in (4.7), we have for the /I -+ g E2 transition matrix element (AL+2
IIQ IIg,L)=
(2L + 5)“2 (2L + 1)
L + 2) Jlr(&, Jw2
&wp, x
s
d/? sin /I df;l,,(/?)(-1
x 1
x(LM2-MIL+20)
‘q
bN- lb’ + bNs2
1
5;” b,vb-,p
Q-Me-iBJy(bt)N
m#O
I-).
(4.14)
The subscript v distinguishes among the m-intrinsic states and will be suppressed for convenience. The intrinsic matrix element in (4.14) is given by (4@l Q-,emiBJy
&J=N!ZN-2(-)M~q~l
(jOi-M12-M) 3
i
(N-l)Y+(N-2)Cr,~)x,x,+Zx;x,]d:,(8) m
+2 1 (-1”enym(jm
I-M-ml2-M)xj,,,x,d’,+,,
m
cn}.
(4.15) Substituting (4.15) in (4.14) and performing approximation, we obtain
the b-integration
using the gaussian
106
KUYUCAK
x
(N-
AND
l)
MORRISON
XjX,C
XkX;
C (JO
k
+
2,
tN-
xjxl
k0
In>)’
F(f2,
I)
I
C(-)" m
5;
x 1 xkxkmxk’xkGm
1 (km k’-m
kk’
IJ,O)
JII
x (k0 k’0 jJIO)(JO
xF((r,,
JIO IZO)2
Z)+xj’xf(f,,
I)
1
+2x(-)“t~(jmL+2OIJm) m XX~~X,~XkXkm~~krnJ-m~lO~ k
I
x (k0 JO IZO) F(T,,
I) .
(4.16)
Due to the orthogonality of the C- G coefficients or x .x’ = 0, leading contributions from all terms (except third) in (4.16) vanish. Also, in the second and fourth terms, the sums over the C- G coefficients effectively bring in factors of l/iV’. Thus, only the m = 1 terms make a contribution to the first layer. Carrying out the summations in (4.16) with the help of Appendix B and dividing by the normalizations (3.10) and (3.36), we obtain (fi, L+211 Q IIg, L)=4i?t(LO20
- 2
IL+20)
)$+ ...}. Ji;cYG~b,J(L)
(4.17)
where B’ = c,, qj,xjx, and B; = cj, (j- !) qi,x,!x,. It appears from (4.17) that there is an extra leading order contribution besides B’ which is what one would obtain in the intrinsic state formalism without projection. In fact, demanding that the ground- and /?-bands be orthogonal gives for 5;” (4.18) and the extra term in (4.17) vanishes. In the consistent Q formalism,
B’= 0, B; =
l/N EXPANSION
IN
107
IBM
(A - I’)b, and manipulating the last term in (4.17) using (3.53)-(3.55) and (4.18), we obtain for the B(E2) value B(E2;L+28-tL,)=a;
3(L+l)(L+2) L2 2(2L + 3)(2L + 5) ??
n-n2+
Y
1
. . . 2.
(4.19)
Equation (4.19) vanishes in the W(3) limit and also in the sd-boson model. Thus, one must go to the second layer terms for a non-zero contribution in the &-model. Observe that the second term in the P-band intrinsic state is essential in getting the right answer, although it does not directly contribute to the final result. This should be contrasted with the calculation without projection where the first term is sufficient to determine the leading contribution to the matrix element. Next, we consider the y + g E2 transition. Replacing the b-band intrinsic state in (4.14) with that of the y-band, we have for the reduced matrix element (2L + 5p2 (2L + 1) (y~L+2”Q”g~L)=[2~(ii,L+2)~($~,L)]1/2M x
s
dflsinpdL-,+2,,
+ bN-’
1
C(LM+22-lqL,+22)
(P) <-I p2
1
<;, bn,,b2-,,,,, Q-,e-iPJ’$b+)N
m # 0,2
I-).
(4.20)
The matrix element in (4.20) is given by (4,’ Q-Me-‘PJy
~~g)=N!ZN-2(-)M~qj, iI x (N-l)Y,+ [ +2(321-M-2
{
(jOI-M12-M)
~~:ltY,-,]xjx,d:,(P) m 12-M)x,zx,d’M+20(fi)
+ 1 tt(-)” (Y,
+ Y2-,(j2-mfm’12-M)x,,_,)x,d!,,,(~)
(4.21)
The rest of the calculation is similar to the previous one. Only the m = 1 terms in (4.21) contribute to the first layer, and the first two terms cancel out due to the orthogonality condition of the ground- and y-bands b, + 2 1 tfvb:” = 0.
(4.22)
108
KUYUCAK
The remaining
AND
MORRISON
terms give
Q llg,L)=J%(LO22\L+22) 28(2)-B;+2C5:“~,“~(1,) Y
) &+
$4.23)
where B; = cjl fl(~2 I- 1 121) qj,x,2xl. In the consistent Q formalism B(2) = n,, B; = 2x2 - (,,/5/2)(2/2y) 6, b,,(b,, = XI (t- 2)112x/~x,~), and the last term in (4.23) is -(,,/?/2)(1/2~)6,. Since b, =b,b,, (this is most easily seen by using the completeness relation in the ground-y-band overlap), only the first term in (4.23) contributes to the B(E2) value, B(E2; L + 2, -+ Lg) = a: A similar calculation to leading order,
(L+3)(L+4) 2(2L+3)(2L+5)
for the y -+ b E2 transition
(y, L+2)1 Q IIp,L>=,:52(LO22
IL+22)
N[n2+
,’ ”
(4.24)
’
gives the following matrix element
B’(2)-$B(Z)+
where B’(2) = B(2, x’, x1). In the consistent Q formalism, (L+3)(L+4) B(E2; L + 2, --t Lp) = a: 2(2L + 3)(2L + 5)
+2y
. ..I.
(4.25)
1
(4.26)
we obtain dn
2
+.
.
. 2.
Due to the complexity of the calculation, we have ignored possible contributions from the higher terms in the /3- and y-band intrinsic operators. However, comparisons with numerical calculations show that Eq. (4.26) is still a good approximation. In the W(3) limit, nz = 0, 7~; = $@, and Eqs. (4.24) and (4.26) reproduce the SU(3) results. Intrinsic state calculation without projection gives the first terms in Eqs. (4.24) and (4.26). Thus the leading order equivalence of the intrinsic state matrix elements with the projected ones breaks down in the case of y -+ /l transition (and, in general, for transitions not involving the ground-band). In such cases, projection should be used in order to obtain reliable results. 4.4. Electromagnetic Excitation of K-Bands Rather than dealing with each K-band separately, we give a single derivation the g -+ K-band excitations via the electromagnetic operator
for
(4.27)
l/h’
EXPANSION
109
IN IBM
which is electric for even K and magnetic for odd K. Note that ti = qjr and tj, = S,, g,. The transition matrix element from (2.7) and (3.42) is
(2L+2K+
2CJ’“(d,m
1)“‘(2L+
1)
L + K) J’-(&, L)l”’
xC(LM+KK-MIL+KK)~~B~~~B~L,+~~(B) m x(-I [
bN-’ b, + bNe2
1
1 tib,b,_,,, rn#O,K
TFke-‘B-‘y(bt)N
I-). (4.28)
The intrinsic matrix element in (4.28) is given by (cjKI T’$cipJ~
I&)=N!
ZN-2(-)Mx
t;
(jOl-M
JK-M)
9 x (N-l)Y,+ [
~~S:u,y,-,]xjx,dtO(B) m
+(-)“Z(jKl
-M-K(K-M)XjECx,d~+KO(B)
+ C (-)”
lm’ (K -M)
JK -M)
YmXjm + (-)”
Y,_,x,,~,]
x,dL.,(P)
.
(4.29)
The first two terms in (4.29) cancel out due to the orthogonality condition, and the last term is supressed through the sums over the C- G coefficients. Thus only the third term contributes to the leading order. Substituting it in (4.28), we obtain after performing the usual steps (K,L+KIl
T()Ilg,L)=[2-6xo]1’2fiE(LOKK)L+KK) X
which gives the following transition
1 t,: (jK 10 (KK) il
strength to leading order,
(2L)! (L + 2K)! B(K; L, -+ (L + x)x) = N2 - 6m) L, (2L + 2K)! Of particular
interest here are the B(M) B(Ml;L,-+(L+l),)=N
(4.30)
XjKX,t,
c tj’X.S 10IKK) 3
xjkx/
1
2. (4.31)
values given by 2::LL+:l1
[T g,x,xll ii]‘.
(4.32)
110
KUYUCAK
AND
MORRISON
Observe that for the spurious l+ state (3.54), Eq. (4.32) does not vanish when TL”’ ) = L, (i.e., g, = rfi), which is another demonstration of the unphysical character of this state. (In fact, 1@i,) = L+/fi I#,),) In the &g-model, we have for the physical K= l-band (L+2)
mfl;L,+(L+l),)=N
2y(2L + 1) x:x:cfi
which clearly vanishes for Trl)
g2
- g‘J2>
(4.33)
= L,.
4.5. g - Factor Variations As noted in Subsection 3.2, stretching of the boson system is unlikely to be observed in energies and E2 transitions. This leaves the Ml properties and in particular g-factors as the most likely candidates. Here, we evaluate the g-factors of the ground-band members. The reduced matrix element of the Ml operator is (Lll 7fM1) l/L) = 2N; x
s
L)c (LMl LT’ M
-Mb-)
d/3 sin B dh,,(b)(-l
bNTY~)e-@‘~(bt)“’
I-).
(4.34)
The matrix element in (4.34) is given by (b,I
TbJ’h’e ~ jpJ, 14,) = NN! ZN-‘(-)M xc
(101 -A411 -M)
g,x;dh,,.
(4.35)
I Substituting
in (4.34) and evaluating (L/l flM1) l/L) = -N
the P-integral, (2L+l)J? F(T L) 3
we obtain c (10 LO IJO)2 IJ
(4.36) The leading term in (4.36) vanishes; hence contrary to quadrupole moments (4.4), the first layer terms here are influenced by the gaussian approximation and are not exact. Evaluating the J-sums with the help of Appendix B and dividing by F(r, L) give
l/N
EXPANSION
IN
111
IBM
With T(l’) = L, in (4.37), the leading term gives 2 fi as it should, but the higher order terms fail to vanish, reflecting the approximation used. In turn, this could be used to estimate the quality of the approximation. For example in the SU(3) limit, the term -2/3 (which is the only term affected by the approximation) should be 2 in order to ensure the vanishing of the higher order terms. The g-factors are defined as g(L) = (Lll FM1) IIL)/Z
Jz.
(4.38)
Clearly, there can be no L dependence for the g-factors in the s&model. &g-model, we have from (4.37) and (4.38) g(L)=g+g$Qf$yl+(~+&)($-2y)+
. ..I.
In the
(4.39)
where g= gJ@ and g’= -& g2 + g,. Substituting (3.24) for the structure coefficients {x,}, we obtain the following expression for the g-factor variation,
4.6. Discussion of Electromagnetic Transitions The l/N expansion for the intraband electromagnetic properties is similar to that for the energies; the leading term CnO is the same or all bands dominated by the g-band intrinsic operator, and C,, provides a l/N correction which distinguishes between different bands. Unfortunately, the B(E2) values are not as well known as the excitation energies, so it is not easy to check whether the Co, terms are correlated as required by the consistent Q formalism. The C,O term provides the socalled boson cutoff. Again, as yet, there are not enough accurate measurements of B(E2) values of high spin states to confirm this effect systematically. In contrast, the interband E2 transitions have varying N-dependence. Each different intrinsic operator between two bands costs a factor of N, and transitions between bands having the same K-quantum number are further surpressed by l/N* in the consistent Q formalism. Thus the N-dependences of the B(E2) values for the transitions, g + g, y --* g, B + y, and fl-+ g, are respectively N2, N, 1, and l/N, which seems to be consistent with the empirical situation [7]. Next, we discuss the Ml properties. The K= l-band is usually considered as a neutron-proton asymmetric state in IBM-2. In the sd-IBM-2 the energy of the K = 3-band is higher than that for the K = l-band, contrary to the experimental situation (E3 N E,/2). Thus inclusion of g-boson in IBM-2 also seems to be necessary. Currently it is not known whether the 1 + states are n-p symmetric or asymmetric because the B(M1; 0 + + 1 + ) values are similar. It may be worthwile to pursue this question further. Another Ml property which may be linked to g-boson is the g-factor variation
112
KUYUCAKANDMORRISON
observed in the g-band. In Eq. (4.40), the first term comes from the l/N expansion of the matrix element and varies smoothly with the quadrupole parameters, while the second term is due to the stretching of the structure coefficients and shows a rapid variation. Although Eq. (4.40) has the potential to explain the whole effect [S], in the parameter range restricted by the energy systematics, it has smaller values and only partially accounts for the data.
5. CONCLUSIONS We have presented an algebraic method for solving IBM hamiltonians. The method is based on angular momentum projection in the intrinsic state formalism and leads to l/N expansion of physical quantities. It has the further advantage that it can be easily generalized to arbitrary kinds of bosons (in fact, the general case is much easier than the specific ones). We have demonstrated the method in solving a general quadrupole hamiltonian, appropriate for the deformed region, and studied the effects of g-boson in energies and electromagnetic transitions. The main contributions of g-boson can be summarized as follows: (i) introduces new bands which are needed experimentally (especially the K = 3-band which cannot be accommodated in the &IBM), (ii) corrects the moment of inertia versus deformation systematics, and (iii) causes stretching in the structure coefficients which leads to g-factor variations among band members. As stressed in Subsection 3.2, the stretching property is unlikely to be observed in quantities related to the quadrupole operator. Ideally, the leading term should be independent of L, which leaves the g-factors as best candidates. Although the present work is confined to ground- and single-phonon excited K-bands, extension to multi-phonon bands is straightforward. The increased complexity of the calculations can be avoided by restricting the expansion to the C,, term. This provides the excitation energy which is the main interest in these bands. The effects of the one-body and other multipole terms in the hamiltonian can be easily studied in perturbation theory when the intrinsic states are still determined by the quadrupole piece. In the general case, the whole hamiltonian enters the variation, and calculation of the structure coefficients is more involved. The energy expressions are modified by the addition of extra terms; however, electromagnetic transitions are still given by the same expressions as presented in Section 4. Previously, the intrinsic state formalism was used in estimating electromagnetic transition rates in deformed nuclei [ 111. Although the formalism is capable of giving answers in general cases, these calculations were mostly restricted to the SU(3) limit. Over the years, a large calculational machinery has been developed for the SU(3) group which has created a theoretical prejudice in favour, in spite of the experimental bias against it. We hope we have made it clear that there is nothing special about the SU(3) limit in the intrinsic state formalism. Based on a simple commutator condition, less restrictive than enforcing a Lie algebra, we have derived
l/NEXPANSION
113
IN IBM
expressions for excitation energies and transition strengths in algebraic form suitable for systematic analysis. With this freedom, it is expected that experiment, rather than group theory, will dictate the quadrupole parameters.
APPENDIX
A: EVALUATION
OF THE /&INTEGRALS
The integral
(A.11
F(r, L, M) = jcu dfi sin /I d&M(&pp2’r 0
was evaluated in Ref. [ 141 for the special case of M = 0 when the d-matrices reduce to the Legendre polynomials. Here, we give a general derivation. dkM(fi) is given by L-M G,,,(P)=
1 t=0
t-1’
(L+M)! (L-M)! (L+M-t)! (L-M-t)!
(t)!2
(cos~~L-2’(sinD”.
(A.2)
Since it involves even powers of cos(fl/2) and sin@/2), it can also be expanded in cosine series. In turn, the product sin(b) dkM can be expanded as sin(B) G,(B)
= f
u”, sin(K+
1)/I.
(A.31
K=O
Substituting thus
(A.3) in (A.1 ), one finds that the resulting integral is standard [ 151;
F(r, L, M) = i K=O
“+I w+
u2”+l
(A.41
(2v+l)!!
The quantities CK aL,(K + 1 )*” + ’ can be obtained by differentiating times with respect to fl for /I = 0. The final result is
(A.3) (2v + 1)
F(T, L, M)=;{+M’+;]
I=’ L4+E23+20-2 TL +4!-[ 44 3
808!! +TiL+sn+
..
1 I
. .. + . .. .
(AS)
114
KUYUCAK
AND
h4ORRISON
Recalling that r= 2/yN, Eq. (A.5) constitutes the l/N expansion which enters all the matrix elements. In order to demonstrate that the l/N expansion is not a result of the gaussian approximation, and also to check on its accuracy, we will evaluate the /?-integrals exactly in some special cases. In the W(3) limit, Z(p)= (cos~)~, where p is the angular momentum of the highest boson. The p-integral for M = 0 is then [ 15 J
=(pN-Lx)!!
2
lPN i
(PN)! (pN+L+
l)!!’ [L* + 1OL + 81
+NrL+21+&
- 3r c2;N)3 [I3 + 28E2 + 108t + 48]+
. . -).
(A.61
Comparing (A.6) with (A.5) in the W(3) limit (r=2/pN), and with M=O, shows that only the first term of each power of l/N is correct in (A.5). In the sd-boson model, Z(p) = a + b cos’ /?, where a = 1 - 3x22, b = 3x:/2. Using the binomial expansion and (A.6), the integral becomes
+&JL’tIOf+8]-&[L+28E2+108t+48]+
..(A-7)
The sums over n can be evaluated by repeatedly integrating x=--,
the binomial
expansion
b a
(A.8)
and expanding the resulting factors of l/(n + k). The final result is Fs&KL)=;
1
1-$L+2(y-l),+&
+8(y2-3y+3),-&
[IL2 + (12y - 14)L [L3 + 4(9y - 1 l)L2
+4(42y2-126y+lll)t+48(y3-7y2+18y-15)]+
...
.
In the SU(3) limit, y = 2 and (A.9) coincides with (A.6) for p = 2. Comparison
(A.91 with
1jNEXPANSION
115
IN IBM
(AS) shows that the gaussian approximation fares worst in the SU(3) limit when y is at a maximum, and becomes better as one moves away from this limit. Thus Eqs. (AS) and (A.6) complement each other.
APPENDIX
B: ANGULAR
MOMENTUM
SUMS
Here we collect the angular momentum sums which are often used in the text. In deriving these, we make use of the identity (I)K(LML'M')zM+M')=(LML'M'~[L*+L'2 +2LoL&+L+L'+L_L'+yIzM+M'),
(B.1)
which gives in some specific cases I(LOL'0
IZO)= [L+L'](LoL'o
IZO)
+ [LL’]“‘((Ll (I)' (LOL'O
L'-
1 IZO) + (L-
IZO)= [E2+4LL'+Lr*](LoL'0 +2[Lt’p2
1 L’l IZO)),
(B.2a)
IZO)
[L+L’-
l]((Ll
L'-
1 IZO) + (L-
1 L’l IIO))
+[L(L-2)t'(z'-2)]"2((L2L'-2~m)+(L-2L'2~zo)), (B.2b) (Q3 (LOL'O
Irn)
= [E3+9L~L2$(9L~2-4L~)E+ + [3L2 + (9L’X((LI
~'-1
P](LOLIO
Irn)
S)L+ 3L” - SE’+ 8][LL’]“’ ~zo)+(~-i~~i
Im))+[3L+3L'-io]
x[L(L-2)L,(L'-2)]"2((L2L'-2(zO) +(L-2L'2(ZO))+[L(L-2)(L-6) xL'(L'-~)@'-~)]"~((L~ The angular momentum
L'-3
IZO) + (L-3
L'3 IZO)).
(B.2c)
sums S,(L, L')=C
(F)* (LO L'O IZO)"
(B.3)
are simply given by the first entries in (B.2), and by S,( L, L') = z4 + 16L’z3 + (36L” - 2OL’)L’ + ( 16L’3 - 20L’2 + 16t’)L 595/181/l-S
+ L14.
(B.4)
116
KUYUCAKANDMORRISON
Another sum involving
6 - j symbols, (B.5)
can be evaluated employing
(B.2) as (B.6a) (-)K J+S---+L-J-K)(L
x(JOL0
1 + 2&Kx (2”
2)
IKO)(J'OL'O
_
-
-,
-, -J-K)
-
1
IKO),
(B.6b)
(L2-((J1+4JK+g2)-2(J+R-1)(C-J-R))
1
-(J’*+4PR+R”)-2(f’+Ki)(L’-J’-R))
x(JOLO~KO)(J'OL'OJKO).
APPENDIX
(B.6c)
C: SU(3) RESULTS
The W(3) limit of the IBM can be solved exactly using group theoretical techniques, and hence provides a valuable check on the calculations. Here, we collect some of the relevant results. The bands used in the text are associated with the following SU( 3) representations, g (pN,O)
P7 Y W-4,2)
K=1,3 W-6,3)
P',f,K=4, (W-894)
whose energies are given by the eigenvalues of the Casimir operator c(~,~)=~2+c12+~~+33(~+~). The vanishing of B(m), m ~0, in the SU(3) limit SU(3) 3 SU(2) x U(1) basis denoted by I.&v),
(C.1) is most easily shown in the
B(m)=c <.W10Pm>q,lx,,-~l=(-I LQAt I->. il
(C.2)
l/NEXPANSION
Since bt I-> = 12p, 0, O), bk I-> =C,
117
IN IBM
D$,,2,c,,,2,(0, n/2, n/2) 12p- 3m, mj2, v> and
Q2 = ($/2)[2A, k i(A + + A_)], Q2 cannot connect the two states and B(2) = 0. For B(l), explicit evaluation gives
(A,, +AZ, +iL& +AZ,,)) l&h 0, 0)
= -$Jj
[l +i-i(l
-i)]=O.
cc.31
In the sd-boson model, the quadrupole parameters take the values, qo2 = 1 and The structure coefficients and their respective eigenvalues are given -l/J?), A’= -1jJ2; x,, = 1, A,$ = by x = (lJ,,& &3), A = ,,/?; x1=(&p, 1/2~;~,=1,1~=-1/~.Otherderivedquantitiesarey=2,a,=4E.,o,~=121, o2 = 241, and p = 712. In the &g-model, the quadrupole parameters are qo2 = 1, qz2 = -( 11/14) ,/‘$, The structure coefficients and their eigenvalues qz4 = 917, and q44 = - (3/14) fi. are given by qz2 = --G/2.
Other derived quantities are y = 4, CT~= 8A, CJ,~= 76A, c’2 = 1121, and p = 55/14.
118
KUYUCAKANDMORRISON
APPENDIX
D: PROJECTED MATRIX ELEMENTS
In this appendix, we give a derivation of Eq. (2.12) for the matrix element of an operator 7’2) between projected intrinsic states, (K’L’M’( Substituting
= (qSK’ 1P $s T;)PhK
T$) (KLM)
the projection
14K).
operator (2.8) and using PgK = Pi*, da’ dQ 0$&Q’) x (4~1 W-2’) T!i?R(Q)
P.1) we get
DLM;((Q) 03.2)
IdK).
Since the aim is to combine the rotation operators, we insert R-‘(Q) Tjyl), and introduce R(i2”) = R(Q) R(O). Rotation of Ti’ is given by R(U)
FN” R-‘(W)
=c D&&2’)
R(B’)
T;!.
after
(D.3)
N’
Eliminating
the remaining
51 dependent piece using (D.4)
and integrating (K’L’M’I
over Q’, we obtain from (D.2) T(NI’ IKLM)
=L8;2 MTN, (LM”IN’ x
s
dQ”D~&T’)(q4,.
(L’K)(LMIN 1 T$R(Q”)
Using the Wigner-Eckart theorem and writing the D-matrice we have for the reduced matrix element (KL’)I
T’l’ \lKL) = ;f2 z
M&w (LM”
IL’M’) JtiK).
and R(W)
(D.5) explicitly,
IN’ )L’K’ > 1 dct d/? dy
x sin Bei(aM”fyK) dh,,K(P)(dK’)
T$e-‘“‘~e-‘fi-‘ve-‘YJ~
J#K). (D.6)
The integrals over c1and y simply give 4n2 and the result (2.12) follows from (D.6) upon dividing by the normalizations.
l/NEXPANSION
lN IBM
119
REPBRENCE~ 1. A. ARIMA AND F. IACHELLQ Adv. Nucl. Phys. 13 (1984), 1. 2. 0. SCHOLTEN, Computer Code PHINT, University of Groningen, The Netherlands, 1976; I. MORRISON, Computer Code SDGBOSON, University of Melbourne, Australia, 1986. 3. N. YOSHINAGA, A. ARIMA, AND T. OTSUKA, Phys. L&f. B 143 (1984) 5. 4. N. YOSHINAGA, Y. AKIYAMA, AND A. ARIMA, Phys. Rev. L.&r. 56 (1986) 1116. 5. S. KUYIJCAK AND I. MORRISON, Phys. Rev. Lat. 58 (1987), 315. 6. R. F. CASTEN AND D. D. WARNER, Prog. Part. Nucl. Phys. 9 (1983), 311. 7. D. D. WARNER AND R. F. CASTEN, Phys. Rev. C 28 (1983), 1798. 8. S. KUYUCAK AND I. MORRISON, Phys. Rev. C 36 (1987), 774. 9. A. ARIMA AND F. IACHELU), Ann. Phys. (N.Y.) 111 (1978), 201. 10. A. E. L. DIEPERINK AND 0. SCHOLTEN, Nucl. Phys. A 346 (1980) 125; J. N. GINNOCHIO AND M. W. KIRSON, Nucl. Phys. A 350 (1980) 31. 11. R. BIJKER AND A. E. L. DIEPERINK, Phys. Rev. C 25 (1982) 2688; H. C. WV, A. E. L. DIEPERINK, AND S. PITTEL, Phys. Rev. C 34 (1986), 703; N. YOSHINAGA, Nucl. Phys. A 456 (1986), 21. 12. J. DUKELSKY, G. G. DUSSEL, R. P. J. PERAZZO, AND H. M. SOFIA, Phys. Left. B 130 (1983) 123; M. C. CAMBIAGGIO, J. DUKELSKY, AND G. R. ZEMBA, Phys. Letr. B 162 (1985) 203. 13. T. OTSUKA AND N. YOSHINAGA, Phys. Left. B 168 (1986) 1. 14. H. A. LAMME AND E. BOEKER, Nucl. Phys. A 111 (1968) 492. 15. I. S. GRADSHTEYN AND I. M. RYZHIK, “Tables of Integrals, Series and Products,” Academic Press, New York. 1980.