- Email: [email protected]
A Generalized pair field approach to the shell model
Nuclear Physics A570 (1994) 377c-386~ North-Holland,Amsterdam
NUCLEAR PHYSICS A
A Generalized Pair Field Approach to the Shell Model Cheng-Li WU and Hsi-Tseng Chen ‘* 8Physics Department, Chung Yuan Christian University Chungli, Taiwan , Republic of China A new approach to the nuclear shell model named as the generalized pair field method (GPFM) is proposed. This method using the optimum pair concept reduces the shell model eigen problems to solving a set of coupled dispersion equations which is tractable even for heavy nuclei. Thus the desired goal of avoiding the dimension difficulty of the traditional shell model approach may be possible. The method can also be applied to a boson system. 1. Introduction
The nuclear shell model is the most well-established microscopic theory in the nuclear structure physics [l]. It has been very successful in explaining nuclear structure at low energy in light nuclei . However, for medium-heavy and heavy nuclei, especially for those with a large number of valence particles, the requisite configuration space is so large that diagonalization of the energy matrix is beyond the capability of the best computer presently available. One possibility for solving this difficulty is to find a new principle for further truncating the shell model space down to a tractable level. This direction was pioneered by the Elliott SU(3) Model [2], in which the concept of dynamical symmetry was the first time introduced to nuclear structure. We may call this as the symmetry ditacted truncation. The Elliott SU(3) model applies only to the s-d shell. This idea was further developed by the Interacting Boson Model (IBM) [3], where the shell model space was drastically truncated to an S-D pair space with fermion pairs being approximated as bosons. Finally, the Fermion Dynamical Symmetry Model ( FDSM ), that we have developed in the late 80’s [4], is able to find such a truncation in the fermionic shell model space without boson approximation, thus extend the traditional shell model into medium-heavy and heavy regions. My colleague professor Guidry will present a talk on the recent progress of this model. Very recently, we with Da Hsuan Feng have found another possibility of solving the shell model difficulty. This approach does not depend much on space truncation but bypasses the usual diagonalization of an energy matrix. We discovered that this is possible after discovering that a system’s many-body wave function is possible to be approximately described by a direct product of pairs with proper structure and spins. The resultant pairs are called optimum pairs, and their existence has been proven and reported in our recent short communication [S]. Although this is still in a very preliminary stage, it *Researchreportedhere are supportedby the NationalScienceCouncilof the Republicof China under the contract NSC 820208M-03%022 03759474/94/$07.CNlO 1994 - Elsevier Science B.V. All rights resewed.
SSDI 037%9474(94)00114-3
378~
C.-L. Wu, H.-T. Chen I A GPF approach to the shell model
looks very promising. It is the purpose of this talk to elaborate on such optimum pair descriptions of the shell model and to present an exposition on the general pair field method that we proposed. 2. The Existence of Optimum Pairs It is well known that the formal solution of the S&Clinger written as
equation for a pair can be
where V is the interaction, EX~the singleparticle energies, and AllXtl\*the pair creation operators. Therefore a pair can be generally defined as
The optimum pairs are defined as those pairs, whose amplitudes Q(XiXsAi) and energies Ed being chosen such that their product is the optimum approximation to the many-body solution of the Schijdinger equation. Namely
(3) where Cl stands for the intermediate angular momentum coupling, and i is the particle index. The question is whether such optimum pairs exist and how good the approximation? One well known example of the existence of the optimum pairs is the pairing model. In such extreme case the optimum pairs are the S pairs. Taking the seniority state ]ZI> ( S]V >= 0) as vacuum, the many-body wave function in this case is just (St)Ns]w >, where Ns is the number of 5’ pairs [5]. In ref. [6] we have shown another example, the Q. Q case. As another extreme, the many-body wave function in this case is generally a strong mixing of different sonorities, or a strong mixing of different pair configurations if the seniority scheme or the low-spin pairs are chosen to be the basis, and yet it becomes a simple direct product of pairs, if the highest-spin pairs are chosen to be the basis. Table 1 presents a shell model wave function analysis for four identical particles moving in a single-j shell. One sees that the shell-model wave functions that shows strong mixing in the low-spin pair basis turn out to manifest themselves in an extremely simple picture: the three low-lying 2+ states are essentislly of pure configuration: ]20,20; 2 > for the first 2+ state, 118,20;2 > for the second 2+ state, and ]18,18; 2 > for the third 2+ state, etc. Only for the high excited state does it start to have significant pair configuration mixing. In other words, the low-spin pair components of these eigenfunctions in the low-spin basis are summed up almost exactly to a single high-spin pair component. It also happens that this desirable feature persists for all the states of the ground, beta and gamma bands, and even for other higher excited bands although they are not as impressive as the lower
C.-L. Wu, H.-T. Chen I A GPF approach to the shell model
379c
bands. The analysis shows that for the present example, these bands can be labeled as (20-20), (B-20), (B-18), . .. . in view of the members of each band being dominantly the states of 120,20; J >, 118,20; J >, [18,18; J >, etc. The whole spectrum is shown in part aofFig. 1.
Table 1 Pair component analyses of J=2 shell model wave functions ( j = 21/2, H = -&a Q , energies are in arbitrary unit) ‘Ihe J=2 shell modelwavefunctionsin high-spinpairbasis 16-16
16-18
-1.45
E
14-16 0.00
0.00
0.00
18-18 0.04
18-20 0.01
1.41
-0.86
-0.01
0.00
-0.13
-0.01
-1.99
0.01
-0.35
0.06
0.20
0.08
-0.08
0.63
0.16
1.79
0.24
0.84
0.98
0.47
-1.49
1.10
20-20
1.38
-0.02
-0.04
-0.13
-0.13
-0.77
-0.17
0.03
0.00 0.00
0.41
-0.06
-0.01
0.00
TheJ=2shellmodel wave functions in low- spinpairbasis
L
2-2
2-4
4-4
4-6
6-6
0.23 0.95
0.33 0.08
0.37 0.97
-0.04 -0.05
0.39 0.72
-0.07
0.62
0.30
0.93
0.21
-0.55
-0.08
-0.86
-0.08
0.93
0.89
1.41
1.02 0.70
0.24
-0.42
-O.48
-1.51
0.29
0.93
-0.05
0.47
-0.23
0.36
1.57
-0.83
0.12
0.23
1 E 1 O-2 -1.45 -0.86 -0.35
-0.53
3. The GPFM in A Single-j Shell After showing the existence of optimum pairs in two extreme cases: the 5’ pairs for monopole pairing and the highest-spin pairs for the &a& interactions, one may ask whether this is true in general? and how the optimum pairs can be obtained dynamically? To answer these questions, a generalized pair field method (GPFM) is introduced which we will now discuss. We begin by considering the example of Cparticle system in a single-j case. Fq. (2) in this situation reduces to
The Hamiltonian can be written in the following form: H =
.zj,/G
(a; x &)” + f c ~~CJ(A$ J
x AJ)”
C.J=< jjJlVljjJ
>
(5)
C.-L. Wu, H.-T. Chen / A GPF approach to the shell model
380~
D
0.2 -
-10
a
0 --
b
b
s’$ -0.2 --
(14-20)
-0.4 -a
b
-10 -9 -8
-0.6 --
-0.8 -a -1
--
b
(18-20)
Figure 1. The spectrum of 4 particles in a single-j shell (j = 2112, H = -Q . Q , energies are in arbitrary unit). (a) The shell model calculation; (b) The GPFM calculation.
-10
-1.2 --
-1.4 --
-1.6 A.
The wave function
-4
=I! (20-20,
is assumed to be
(6) Introducing H’=H-wxIij
an auxiliary
Hamiltonian
with a multiplier
w (7)
i
where Iii stands for the identity interaction between i’s and j’s particles. Obviously, H’ and H have the same eigenfunctions, and the eigen-energies only differ by a constant,
Thus, solving eigen-problem of H is equivalent to solve that of H’. With the proper choice of w , it will turn the resulting wave function into the desired form of the optimum pairs. Applying H’ to Pq. (6)
H/I’@,) =
+
(~3 +
8,
E2)1'@J)
(9)
C.-L. Wu, H.-T. Chen I A GPF approach to the shell model
381c
where kiz and A are defined as follows: hz
=
)(CJ,
-(&-
+
c,,
-
2u)~,2J~xo’jJ~;jj&;
J1JzJ)
(10)
12
In Bqs. (lo)- (11) X is a 9 - j symbol and j = sqrt2J + 1. The prime on the sum in Eq. (11) excludes the values Ji Ji = 51Jz and Jz 51. Now we demand the pair energies Ei and Ez to satisfy the following coupled dispersion equations 1+1+-
kis CJI - w 2&j - El +g=Y&= CJ,
-
0
(12)
h2
w
0
2&j - Es +E1=
The multiplier w can be determined by minimizing the X2, i. e. the square sum of the amplitude in A: x2 = JG, ’ [(CJ; + CJ; - 2w)j,j2j:j~XcijJl;jjJ2; 12
J;J;J)]’
(13)
This can be achieved by setting [@X(jjJI;jjJ2; W= f c
‘(CJ;
+ CJ;)W(JIJ~)
WJlJ2)
,
J;J;J)12
=
J’12J’
(14) c;;J;
[j:j:X(jjJI;jjJ2;
J;J;J)12
If A is negligible (i. e. X2 small), which is now in the spirit of an independent pair approximation, Eq. (9) becomes H’IQJ)
=
(El
(15)
+ E~)I~J)
Thus, an independent pair like solution for H’ is obtained: The eigenfunction is a direct product of two individual pairs (Bit x B?,)J 10) , and the eigen-energy is a sum of individual pair energies, which are determined by solving the coupled dispersion equations (Eq. (12)). For the simple 4particle example, the coupled dispersion equations can be easily solved: multiplying the first equation by 2~ - El, and the second equation by 2~ - Es and then summing up, we obtain E’ = (EI +
E2)
=
4E +
(CJ,
+ CJ,
-
2~
+
ku.) ,
C,=x{;
;
:)
(16)
38%
C.-L. Wu, H.-T. Chen f A GPF a~~roa~h to the sheil model
Choosing the high-spin pair basis, inserted CJ, E’ (Eq. (16)), w (Eq. (14)) and k12 (Eq. (10)) into Eq. (8), the exact shell model results for low lying states are very well reproduced. The detail comparison is shown in Fig. l(b). Obviously, the above formulation need not be restricted to just Q - Q inter~tion; it can be apply to any other interaction as well. The key point is to fine the optimum pairs that make A negligible. For Q . Q interaction, the optimum pairs are the highest-spin pairs as have shown; for other interactions, the optimum pairs may be different. In general, the optimum pairs should be determined by choosing those which minimized the x2 (Eq. (13)), which is a measurement of the validity of neglecting A, thus the validity of optimum pair appr~mation. 4. The
in Multi-j
GPFM
Shells
The extension to N pairs moving in a multi-j shell is straightforward. The optimum pairs and the wave functions of a many-body system are defined by Eqs. (2)- (3). Similar to the single-j case, one can obtain
(17)
x ( BX,Bh,...A!,xai\i...B~~)~
IO) + A
where A = A, + A,: AW =
(--I” x 2
x ’ (Bt,B~~...A’12ni...At*~...B~~)~ all indiaa
[ (?jr(121’2’,34)
x
Akj = C ifj
+ q;(341’2’,
C
12341’2’
X [U~?(l21’%34)
& 3
X
((-)’
x
~~(12341’2’,A~A~)~~n(l2341’2’)
(18)
10)
12) - 2~)) ?‘(1234,&A;)]
i
(19) (B~,B~~...Ar,i...A~j,...Bh,);: IO)
+ v:;(341’f,
12) -
w(&,r2
+
&‘2$34)]
- ~~(121f~)C~6(34A~)~
The above equations are valid for both fermionic and bosonic systems with the exchange parity (-)P = - 1 for fermions and + 1 for bosons. The coupled dispersion equations are :
(20)
C.-L. Wu, H.-T. Chen I A GPF approach to the sheI model
383~
Here a short hand notation has been used: the indices 121’2’... stand for X1, X2, X11,X2, ... , and I&s, the interaction matrix elements (XrXzhijVIXr~Xs,h~).The sum over all indices in Eq. (19) includes 12341’2’flfiA:Ai and Q (i # j), and the prime on the sum excludes the values Ai A> = Ai Aj and Aj Ai when Q’ = R. The quantities kj and w can be obtained by minimizing xLj and xz which are the squire sum of the amplitudes in Akj and A, respectively: 121’2’, 34) + v;F’(341’2’,12) - 2~)) e”‘(1234,&A;)]2 J
XE=
X
(~)p~~(12341’2’,A~A~)Z~(12341’6)
(21)
- rC,j(121’2’)Cz8(34Aj)}2
The results are 121’27.34 ) + vy’(341’2’; ,
kj(l21’2’) = (-)“( 1 +I,,)
*,
CWc!(1234,
12)) W,R”‘(1234,1’2’A;A;)
(23)
(24)
l’dhihj)
3.4
+ &,&] x [Vp(121’2’; 34) + vgy341’2’; 12) - w(51,2t,12
$iF (12341’2’, AiAj)
These results are quite similar to that of the singlej case (see Eq. (10) and Eq. (14))) except now more indices to sum up, and that the interaction matrix elements CJ’S and weights W( Jr, J2) in Eq. (10) and Eq. (14) are replaced by a sort of average ones: vf,o’ and wrr for w; Vt,? and WC: for hj. Note that in the multi-shell case kij can only be determined in an average manner with the weights qn, while for the single-j case the weight is one. The average interaction matrix elements and the weights are qr
e”‘( 1’2’A:34A)) (121’2’,34) = v&t pR’(l234 l,2,A!Af,) , ij * 3
Z~*(l’2’Ai34Aj)y~n(l’2’Ai34Aj) V;,?(121’2’,34)= V&t , 12’2 g$* (1234,1’2’)jj;“(1234,l’a’AiAj)
(25)
(26)
(27) ?Wf(1234,1’2’A;A))
=
C.-L. Wu, H.-T. Chen I A GPF approachto the shell model
384c
~‘(1234,1’2%&) zy(1234,1’2’)
=
+ Fjj”‘(1’2’A)12A;)
(29)
2
2;o(1’3,2’4)
=
+ 2?(2’2,1’1)
(30)
2
The quantities Go’ $7’7
1’342’4A;)
2F(
1’312:2’4A;) =
Y~(1’2’A~34A~)
G”‘(1’2’A;34A;)
and 2,;” and y,Go are defined as follows:
= C$&$$jn’(
1’2’A;34A;) ,
c?i _ @(1’3Ai) l 3 - El! + &3 - I$
(31)
e( 1’3AJCz EI,+E3_E2,_bq Y:*W’UW
( 1+
Ej-Ei
= C &oij n-nij
(32)
>
/3,$o, &AjA&X(l’SAi;
2’4A.j; A:A:.sz2)
(33)
where /3$ nl, ‘s are the angular momentum recoupling coefficients: v
cI@% )* R
,fijB!ie (L 1 (A~lkd*! 0
IO>= C DE”,, R”Rlj
’
5-P
’
ALhAj
Rlj
A
lo)
(34)
Thus, if x~j and Xt are small enough, we obtain approximate solutions for the manybody system: the eigen functions are the optimum pair products (Eq. (3)) and the eigen energies are
Both the optimum pair energies Ei and the pair structure Q(XiXzAi)‘s can be determined by solving the coupled dispersion equations Eq. (20). It should be noted that, unlike single-j case, the values of lcij and w now depend on the pair structure 0(XiXsAi)‘s , which are unknown, and have weak energy dependence as well (through 7$“’ and 2Gn, see Eqs. (31)- (32)). Therefore, the coupled dispersion equations Eq. (20) generally should be solved in a self consistent manner. This can be achieved by taking the two-body solutions of the system Hamiltonian as initial values of Q(XiXzAi)‘s and E~‘s to obtain the initial values for kij and W, and then start the iteration. Thus the diagonalization of a huge matrix in usual shell model calculations is now replaced by solving N x Nd coupled dispersion equations, where Nd is the dimension of the two-particle configurations. Although it is still nontrivial for a realistic calculation, but it is tractable even for heavy nuclei. The detail discussions about the extension to the multi-shell will be published in a subsequent publication. We must emphasis that although the present optimum pair picture is akin to an independent pair picture in many aspects, they are really not the same. Unlike the concept of an static common potential well, the pair field here is sensitively particle number and
C.-L. Wu, H.-T. Chen I A GPF approach to the shell model
38%
state dependent, because there are exchange effects in w and &, which are state and number dependent. Only by neglecting w and kij will a common potential emerge and is equivalent to the boson energies in the Tamm-Dancoff approximation. In addition, the choice of optimum pairs (for an n-p system, one may have to include n-p pairs ) and their structnre will also vary from state to state. Thus, the determination of pair mean field in the GPFM is highly non-trivial. For each state the energies and structures of the optimum pairs must be self insistently determined from the coupled dispersion equations. Furthermore, one should note that there is actually a free choice of the auxiliary Hamiltonian and Eq. (7) is merely the simplest one. In fact any scalar, if it is known to be (or approximately) diagonal in the optimum pair basis and has parameters to minimize x2 making A negligible, will be a good choice to replace w Ci
NUCLEAR PHYSICS A
A Generalized Pair Field Approach to the Shell Model Cheng-Li WU and Hsi-Tseng Chen ‘* 8Physics Department, Chung Yuan Christian University Chungli, Taiwan , Republic of China A new approach to the nuclear shell model named as the generalized pair field method (GPFM) is proposed. This method using the optimum pair concept reduces the shell model eigen problems to solving a set of coupled dispersion equations which is tractable even for heavy nuclei. Thus the desired goal of avoiding the dimension difficulty of the traditional shell model approach may be possible. The method can also be applied to a boson system. 1. Introduction
The nuclear shell model is the most well-established microscopic theory in the nuclear structure physics [l]. It has been very successful in explaining nuclear structure at low energy in light nuclei . However, for medium-heavy and heavy nuclei, especially for those with a large number of valence particles, the requisite configuration space is so large that diagonalization of the energy matrix is beyond the capability of the best computer presently available. One possibility for solving this difficulty is to find a new principle for further truncating the shell model space down to a tractable level. This direction was pioneered by the Elliott SU(3) Model [2], in which the concept of dynamical symmetry was the first time introduced to nuclear structure. We may call this as the symmetry ditacted truncation. The Elliott SU(3) model applies only to the s-d shell. This idea was further developed by the Interacting Boson Model (IBM) [3], where the shell model space was drastically truncated to an S-D pair space with fermion pairs being approximated as bosons. Finally, the Fermion Dynamical Symmetry Model ( FDSM ), that we have developed in the late 80’s [4], is able to find such a truncation in the fermionic shell model space without boson approximation, thus extend the traditional shell model into medium-heavy and heavy regions. My colleague professor Guidry will present a talk on the recent progress of this model. Very recently, we with Da Hsuan Feng have found another possibility of solving the shell model difficulty. This approach does not depend much on space truncation but bypasses the usual diagonalization of an energy matrix. We discovered that this is possible after discovering that a system’s many-body wave function is possible to be approximately described by a direct product of pairs with proper structure and spins. The resultant pairs are called optimum pairs, and their existence has been proven and reported in our recent short communication [S]. Although this is still in a very preliminary stage, it *Researchreportedhere are supportedby the NationalScienceCouncilof the Republicof China under the contract NSC 820208M-03%022 03759474/94/$07.CNlO 1994 - Elsevier Science B.V. All rights resewed.
SSDI 037%9474(94)00114-3
378~
C.-L. Wu, H.-T. Chen I A GPF approach to the shell model
looks very promising. It is the purpose of this talk to elaborate on such optimum pair descriptions of the shell model and to present an exposition on the general pair field method that we proposed. 2. The Existence of Optimum Pairs It is well known that the formal solution of the S&Clinger written as
equation for a pair can be
where V is the interaction, EX~the singleparticle energies, and AllXtl\*the pair creation operators. Therefore a pair can be generally defined as
The optimum pairs are defined as those pairs, whose amplitudes Q(XiXsAi) and energies Ed being chosen such that their product is the optimum approximation to the many-body solution of the Schijdinger equation. Namely
(3) where Cl stands for the intermediate angular momentum coupling, and i is the particle index. The question is whether such optimum pairs exist and how good the approximation? One well known example of the existence of the optimum pairs is the pairing model. In such extreme case the optimum pairs are the S pairs. Taking the seniority state ]ZI> ( S]V >= 0) as vacuum, the many-body wave function in this case is just (St)Ns]w >, where Ns is the number of 5’ pairs [5]. In ref. [6] we have shown another example, the Q. Q case. As another extreme, the many-body wave function in this case is generally a strong mixing of different sonorities, or a strong mixing of different pair configurations if the seniority scheme or the low-spin pairs are chosen to be the basis, and yet it becomes a simple direct product of pairs, if the highest-spin pairs are chosen to be the basis. Table 1 presents a shell model wave function analysis for four identical particles moving in a single-j shell. One sees that the shell-model wave functions that shows strong mixing in the low-spin pair basis turn out to manifest themselves in an extremely simple picture: the three low-lying 2+ states are essentislly of pure configuration: ]20,20; 2 > for the first 2+ state, 118,20;2 > for the second 2+ state, and ]18,18; 2 > for the third 2+ state, etc. Only for the high excited state does it start to have significant pair configuration mixing. In other words, the low-spin pair components of these eigenfunctions in the low-spin basis are summed up almost exactly to a single high-spin pair component. It also happens that this desirable feature persists for all the states of the ground, beta and gamma bands, and even for other higher excited bands although they are not as impressive as the lower
C.-L. Wu, H.-T. Chen I A GPF approach to the shell model
379c
bands. The analysis shows that for the present example, these bands can be labeled as (20-20), (B-20), (B-18), . .. . in view of the members of each band being dominantly the states of 120,20; J >, 118,20; J >, [18,18; J >, etc. The whole spectrum is shown in part aofFig. 1.
Table 1 Pair component analyses of J=2 shell model wave functions ( j = 21/2, H = -&a Q , energies are in arbitrary unit) ‘Ihe J=2 shell modelwavefunctionsin high-spinpairbasis 16-16
16-18
-1.45
E
14-16 0.00
0.00
0.00
18-18 0.04
18-20 0.01
1.41
-0.86
-0.01
0.00
-0.13
-0.01
-1.99
0.01
-0.35
0.06
0.20
0.08
-0.08
0.63
0.16
1.79
0.24
0.84
0.98
0.47
-1.49
1.10
20-20
1.38
-0.02
-0.04
-0.13
-0.13
-0.77
-0.17
0.03
0.00 0.00
0.41
-0.06
-0.01
0.00
TheJ=2shellmodel wave functions in low- spinpairbasis
L
2-2
2-4
4-4
4-6
6-6
0.23 0.95
0.33 0.08
0.37 0.97
-0.04 -0.05
0.39 0.72
-0.07
0.62
0.30
0.93
0.21
-0.55
-0.08
-0.86
-0.08
0.93
0.89
1.41
1.02 0.70
0.24
-0.42
-O.48
-1.51
0.29
0.93
-0.05
0.47
-0.23
0.36
1.57
-0.83
0.12
0.23
1 E 1 O-2 -1.45 -0.86 -0.35
-0.53
3. The GPFM in A Single-j Shell After showing the existence of optimum pairs in two extreme cases: the 5’ pairs for monopole pairing and the highest-spin pairs for the &a& interactions, one may ask whether this is true in general? and how the optimum pairs can be obtained dynamically? To answer these questions, a generalized pair field method (GPFM) is introduced which we will now discuss. We begin by considering the example of Cparticle system in a single-j case. Fq. (2) in this situation reduces to
The Hamiltonian can be written in the following form: H =
.zj,/G
(a; x &)” + f c ~~CJ(A$ J
x AJ)”
C.J=< jjJlVljjJ
>
(5)
C.-L. Wu, H.-T. Chen / A GPF approach to the shell model
380~
D
0.2 -
-10
a
0 --
b
b
s’$ -0.2 --
(14-20)
-0.4 -a
b
-10 -9 -8
-0.6 --
-0.8 -a -1
--
b
(18-20)
Figure 1. The spectrum of 4 particles in a single-j shell (j = 2112, H = -Q . Q , energies are in arbitrary unit). (a) The shell model calculation; (b) The GPFM calculation.
-10
-1.2 --
-1.4 --
-1.6 A.
The wave function
-4
=I! (20-20,
is assumed to be
(6) Introducing H’=H-wxIij
an auxiliary
Hamiltonian
with a multiplier
w (7)
i
where Iii stands for the identity interaction between i’s and j’s particles. Obviously, H’ and H have the same eigenfunctions, and the eigen-energies only differ by a constant,
Thus, solving eigen-problem of H is equivalent to solve that of H’. With the proper choice of w , it will turn the resulting wave function into the desired form of the optimum pairs. Applying H’ to Pq. (6)
H/I’@,) =
+
(~3 +
8,
E2)1'@J)
(9)
C.-L. Wu, H.-T. Chen I A GPF approach to the shell model
381c
where kiz and A are defined as follows: hz
=
)(CJ,
-(&-
+
c,,
-
2u)~,2J~xo’jJ~;jj&;
J1JzJ)
(10)
12
In Bqs. (lo)- (11) X is a 9 - j symbol and j = sqrt2J + 1. The prime on the sum in Eq. (11) excludes the values Ji Ji = 51Jz and Jz 51. Now we demand the pair energies Ei and Ez to satisfy the following coupled dispersion equations 1+1+-
kis CJI - w 2&j - El +g=Y&= CJ,
-
0
(12)
h2
w
0
2&j - Es +E1=
The multiplier w can be determined by minimizing the X2, i. e. the square sum of the amplitude in A: x2 = JG, ’ [(CJ; + CJ; - 2w)j,j2j:j~XcijJl;jjJ2; 12
J;J;J)]’
(13)
This can be achieved by setting [@X(jjJI;jjJ2; W= f c
‘(CJ;
+ CJ;)W(JIJ~)
WJlJ2)
,
J;J;J)12
=
J’12J’
(14) c;;J;
[j:j:X(jjJI;jjJ2;
J;J;J)12
If A is negligible (i. e. X2 small), which is now in the spirit of an independent pair approximation, Eq. (9) becomes H’IQJ)
=
(El
(15)
+ E~)I~J)
Thus, an independent pair like solution for H’ is obtained: The eigenfunction is a direct product of two individual pairs (Bit x B?,)J 10) , and the eigen-energy is a sum of individual pair energies, which are determined by solving the coupled dispersion equations (Eq. (12)). For the simple 4particle example, the coupled dispersion equations can be easily solved: multiplying the first equation by 2~ - El, and the second equation by 2~ - Es and then summing up, we obtain E’ = (EI +
E2)
=
4E +
(CJ,
+ CJ,
-
2~
+
ku.) ,
C,=x{;
;
:)
(16)
38%
C.-L. Wu, H.-T. Chen f A GPF a~~roa~h to the sheil model
Choosing the high-spin pair basis, inserted CJ, E’ (Eq. (16)), w (Eq. (14)) and k12 (Eq. (10)) into Eq. (8), the exact shell model results for low lying states are very well reproduced. The detail comparison is shown in Fig. l(b). Obviously, the above formulation need not be restricted to just Q - Q inter~tion; it can be apply to any other interaction as well. The key point is to fine the optimum pairs that make A negligible. For Q . Q interaction, the optimum pairs are the highest-spin pairs as have shown; for other interactions, the optimum pairs may be different. In general, the optimum pairs should be determined by choosing those which minimized the x2 (Eq. (13)), which is a measurement of the validity of neglecting A, thus the validity of optimum pair appr~mation. 4. The
in Multi-j
GPFM
Shells
The extension to N pairs moving in a multi-j shell is straightforward. The optimum pairs and the wave functions of a many-body system are defined by Eqs. (2)- (3). Similar to the single-j case, one can obtain
(17)
x ( BX,Bh,...A!,xai\i...B~~)~
IO) + A
where A = A, + A,: AW =
(--I” x 2
x ’ (Bt,B~~...A’12ni...At*~...B~~)~ all indiaa
[ (?jr(121’2’,34)
x
Akj = C ifj
+ q;(341’2’,
C
12341’2’
X [U~?(l21’%34)
& 3
X
((-)’
x
~~(12341’2’,A~A~)~~n(l2341’2’)
(18)
10)
12) - 2~)) ?‘(1234,&A;)]
i
(19) (B~,B~~...Ar,i...A~j,...Bh,);: IO)
+ v:;(341’f,
12) -
w(&,r2
+
&‘2$34)]
- ~~(121f~)C~6(34A~)~
The above equations are valid for both fermionic and bosonic systems with the exchange parity (-)P = - 1 for fermions and + 1 for bosons. The coupled dispersion equations are :
(20)
C.-L. Wu, H.-T. Chen I A GPF approach to the sheI model
383~
Here a short hand notation has been used: the indices 121’2’... stand for X1, X2, X11,X2, ... , and I&s, the interaction matrix elements (XrXzhijVIXr~Xs,h~).The sum over all indices in Eq. (19) includes 12341’2’flfiA:Ai and Q (i # j), and the prime on the sum excludes the values Ai A> = Ai Aj and Aj Ai when Q’ = R. The quantities kj and w can be obtained by minimizing xLj and xz which are the squire sum of the amplitudes in Akj and A, respectively: 121’2’, 34) + v;F’(341’2’,12) - 2~)) e”‘(1234,&A;)]2 J
XE=
X
(~)p~~(12341’2’,A~A~)Z~(12341’6)
(21)
- rC,j(121’2’)Cz8(34Aj)}2
The results are 121’27.34 ) + vy’(341’2’; ,
kj(l21’2’) = (-)“( 1 +I,,)
*,
CWc!(1234,
12)) W,R”‘(1234,1’2’A;A;)
(23)
(24)
l’dhihj)
3.4
+ &,&] x [Vp(121’2’; 34) + vgy341’2’; 12) - w(51,2t,12
$iF (12341’2’, AiAj)
These results are quite similar to that of the singlej case (see Eq. (10) and Eq. (14))) except now more indices to sum up, and that the interaction matrix elements CJ’S and weights W( Jr, J2) in Eq. (10) and Eq. (14) are replaced by a sort of average ones: vf,o’ and wrr for w; Vt,? and WC: for hj. Note that in the multi-shell case kij can only be determined in an average manner with the weights qn, while for the single-j case the weight is one. The average interaction matrix elements and the weights are qr
e”‘( 1’2’A:34A)) (121’2’,34) = v&t pR’(l234 l,2,A!Af,) , ij * 3
Z~*(l’2’Ai34Aj)y~n(l’2’Ai34Aj) V;,?(121’2’,34)= V&t , 12’2 g$* (1234,1’2’)jj;“(1234,l’a’AiAj)
(25)
(26)
(27) ?Wf(1234,1’2’A;A))
=
C.-L. Wu, H.-T. Chen I A GPF approachto the shell model
384c
~‘(1234,1’2%&) zy(1234,1’2’)
=
+ Fjj”‘(1’2’A)12A;)
(29)
2
2;o(1’3,2’4)
=
+ 2?(2’2,1’1)
(30)
2
The quantities Go’ $7’7
1’342’4A;)
2F(
1’312:2’4A;) =
Y~(1’2’A~34A~)
G”‘(1’2’A;34A;)
and 2,;” and y,Go are defined as follows:
= C$&$$jn’(
1’2’A;34A;) ,
c?i _ @(1’3Ai) l 3 - El! + &3 - I$
(31)
e( 1’3AJCz EI,+E3_E2,_bq Y:*W’UW
( 1+
Ej-Ei
= C &oij n-nij
(32)
>
/3,$o, &AjA&X(l’SAi;
2’4A.j; A:A:.sz2)
(33)
where /3$ nl, ‘s are the angular momentum recoupling coefficients: v
cI@% )* R
,fijB!ie (L 1 (A~lkd*! 0
IO>= C DE”,, R”Rlj
’
5-P
’
ALhAj
Rlj
A
lo)
(34)
Thus, if x~j and Xt are small enough, we obtain approximate solutions for the manybody system: the eigen functions are the optimum pair products (Eq. (3)) and the eigen energies are
Both the optimum pair energies Ei and the pair structure Q(XiXzAi)‘s can be determined by solving the coupled dispersion equations Eq. (20). It should be noted that, unlike single-j case, the values of lcij and w now depend on the pair structure 0(XiXsAi)‘s , which are unknown, and have weak energy dependence as well (through 7$“’ and 2Gn, see Eqs. (31)- (32)). Therefore, the coupled dispersion equations Eq. (20) generally should be solved in a self consistent manner. This can be achieved by taking the two-body solutions of the system Hamiltonian as initial values of Q(XiXzAi)‘s and E~‘s to obtain the initial values for kij and W, and then start the iteration. Thus the diagonalization of a huge matrix in usual shell model calculations is now replaced by solving N x Nd coupled dispersion equations, where Nd is the dimension of the two-particle configurations. Although it is still nontrivial for a realistic calculation, but it is tractable even for heavy nuclei. The detail discussions about the extension to the multi-shell will be published in a subsequent publication. We must emphasis that although the present optimum pair picture is akin to an independent pair picture in many aspects, they are really not the same. Unlike the concept of an static common potential well, the pair field here is sensitively particle number and
C.-L. Wu, H.-T. Chen I A GPF approach to the shell model
38%
state dependent, because there are exchange effects in w and &, which are state and number dependent. Only by neglecting w and kij will a common potential emerge and is equivalent to the boson energies in the Tamm-Dancoff approximation. In addition, the choice of optimum pairs (for an n-p system, one may have to include n-p pairs ) and their structnre will also vary from state to state. Thus, the determination of pair mean field in the GPFM is highly non-trivial. For each state the energies and structures of the optimum pairs must be self insistently determined from the coupled dispersion equations. Furthermore, one should note that there is actually a free choice of the auxiliary Hamiltonian and Eq. (7) is merely the simplest one. In fact any scalar, if it is known to be (or approximately) diagonal in the optimum pair basis and has parameters to minimize x2 making A negligible, will be a good choice to replace w Ci