- Email: [email protected]
Quasiconfigurations: An approach to effective forces
Volume 82B, number 3,4
PHYSICS LETTERS
9 April 1979
QUASICONFIGURATIONS: AN APPROACH TO EFFECTIVE FORCES A. POVES, E. PASQUINI 1 and A.P. ZUKER Laboratoire de Physique Nucldaire Th(orique, CRN, 67037 Strasbourg Cedex, France
Received 7 January 1979
Many-body effective operators appear naturally by dressing states through a perturbative unitary transformation. They have forms that differ from those obtained in the Bloch-Horowitz approach. The fn/2 problem is treated explicitly. Pandya's transforms are generalized.
The success of the shell model can be equated to the possibility of describing properties in the vicinity of the ground states by the interaction between (few) particles confined in a small number of active orbits defined by a (hopefully) self-consistent calculation. For a long time it has been recognized that couplings to levels outside these basic configurations are important, and they have been the subject of extensive studies [1]. It is the purpose o f this paper to show that, provided perturbative methods are valid, such couplings can be accounted for rigorously, by introducing dressed many-particle states (quasiconfigurations). The idea is based on a revival of degenerate perturbation theory as explained in most QM books following Van Vleck's article of 1929 [2] Although it is quite feasible to cast our equations in a general framework, we have decided to present the problem using a notation that allows to visualize clearly the particular case we have dealt with in detail. Consider the f7/2 orbital (region from 40Ca to 56Ni) as spanning a well-defined model space. The validity of this assumption is borne out by classical calculations [3], by experimental fact and by careful SM studies [4]. Before coming to a diagonalization, the fn configurations have to be "dressed" to account for the following phenomena: (a) Realistic forces are always sufficiently singular to impose a G-matrix calculation. Many such calculations exist and have a striking common feature: different potentials fitting the two-nucleon data, subjected to different Brueckner treatments lead to extremely similar G-matrix elements [4]. We adopt for our calculations the classic K u o - B r o w n values [5] and refer to the discussions in ref. [4] for justification of the (only !) important change in monopole behavior. (b) The fp space couples to all possible multipole excitations of the core. The coupling may well be mediated by giant resonances, but it exists whether the latter exist or not. It is our opinion (to be sustained elsewhere) that for effects other than monopole, the bubble diagrams as used in KB [5] are a good approximation. (c) The fp space also couples to intruder states such as the famous 4 p - 2 h excitations in 42Ca for which experimental evidence is overwhelming [6]. Mixing is strong (but local) at the beginning of the shell because of vanishing energy denominators and perturbative treatment breaks down. Later on (after A = 46) the intruders move up and the fairly weak off-diagonal couplings allow to ignore them. As a consequence it is strongly advised to stop bothering about some individual states in the neighbourhood of 4°Ca and consider their fp centroids (well defined because of the locality o f the mixing). 1 Eduardo Pasquini disappeared in Argentina over two years ago and we have had no news of him (and his wife) since. Our results owe so much to his work, that his appearence as cosigner is more than justified. 319
Volume 82B, number 3,4
PHYSICS LETTERS
9 April 1979
(d) Once the preceding steps are followed we have a good fp model space and it only remains to diagonalize matrices ranging up to 10 8-dimensional. To solve the problem we take up again the hints of refs. [3,4] : fn configurations are sufficiently dominant and some perturbative treatment seems warranted. Now, we shall propose a scheme, adapted to the case at hand but capable of full generality through obvious extensions. First we need to say a word on the notation. We shall use throughout second quantization in the coupled representation following French's article [7]. The hamiltonian is
H = ~ei[i]l/2(AiBi) 0 i
~ [I']I/2w~I[(AiA/)r(B~BI)rlO~q~tc I . ijkl,I" i<~j,R<~l
(1)
ijkl run over the orbits f7/2 (from now on f) and f5/2 P3/2 Pl/2 (from now on, generically r). [' = J T i f w e use isospin representation. I" ~-J if we use np representation. [r] = ( 2 J + 1)(2T + 1) (or ( 2 J + 1)). ( _ ) r = (_)J+T (or (--)J)" ~ij = (1 + 8ij)-1/2. A is the usual creation operator a +, B is the tensor constructed out of the annihilation operator a.
B( " "i+iz tz = I.--) ai_iz . The basis for some n-particle level of given spin and isospin P, is given by the set: If n PXn), If n-1 r [,Xn_l), [fn-2r2FXn_2) etc., where X i are the extra quantum members specifying the states. For simplicity, we shall denote the basis by fn, fff-1, f,~-2 etc. At the beginning the hamiltonian spans the whole space. We are interested in decoupling some model block (fn) from the rest of the full (fr) n matrix. The typical matrix element (fnlHI fn-m) is a linear combination of Wfffr (multipole, m = 1) or Wffrr (pairing, m = 2) terms. We say it is of order W (O(I4/)). Now we dress the states fn (and fn-m in general) by introducing the perturbative unitary transformation -nf~ =f~+n
~
#,rn=l,2
(gc~ - -(o)-l(fnlVlfff -m) • tff-m + normalization ,
counterterms to O(V2/g 2) ,
(2)
and similar expressions for -(n-re. The splitting, H = h + V, between unperturbed (h with eigenvalues g) and perturbation (V) terms will be examined later. The following results obtain:
(-(n-mj-(~-m') = 8o~Smm, + O(V3/~-3),
m, m ' = 0, 1 , 2 ,
(3)
(fnIHl-f~-m)= Cm [ ( f n l H l f ~ - m ) + ~=~*(-ge~--g~,)-l(fnlHlf~+m-3)(f~+m-31Hlf~Tm)] + O ( V 3 / g 2 ) ,
(4)
where
Cm = 1 - ~ ( g c ~ ' - - g ~ ' ) - l ( ( f n l H l f n ' ) - ( f ~ ' - m l H l f ~ - m ) ) , ~'/J'
rn = 1 , 2 ,
+ { v, m=~l [(ga -- g3,)-1 + (gt3 - - ' - l " f n ' g ' f n - m ' < f . ~ - m l g l f ~ ) + O ( V 3 / g 3 ) {~-nIglf~ -n > = n = ,2 e v) 1~ c~l I ~, ~
(5)
•
(6)
Eq. (3) insures that we have a good unitary transformation to O(V3/~-3). Eqs. (4) and (5) show that the offdiagonal matrix between the dressed states ~-n and ~-~-rn have been reduced to O(V3/g 2) provided the energy denominators ~-, which we have not defined yet, are chosen to cancel Cm of eq. (5). To understand how this crucial choice is to be made, we first need to work on eq. (6). As it stands, it is Van Vleck's result [2] with one interesting improvement: the unperturbed states fn are not assumed to be degenerate and the energy denominators for the off-diagonal terms are nicely symmetric, insuring the hermiticity of the effective hamiltonian. Now, we are going to take advantage of the appearance, in eq. (6), of complete sets of intermediate states that will allow us to use closure, provided the energy denominators are constant for each complete set. As pairing and multipole excitations contribute in different ways we are going to separate them explicitly. We introduce the notations: 320
Volume 82B, number 3,4 W1 = Wfffr,
PHYSICS LETTERS
eml=~((ea---(Tm)-l+(-(6---(q,m)-l),
W2 = Wffrr ,
9 April 1979 m=l,2.
(7)
Here we have made explicit the m dependence of the intermediate states, while omitting any reference in gm to other quantum numbers. To invoke closure em has to be a constant for each complete set. Now we can separate f and r spaces in eq. (6) using the standard result for the reduced matrix element of coupled spherical tensors and sum over intermediate states. The contribution from r space reduces to a constant. For the pairing term four fermion operators remain (two-body force), while the multipole term has six (two- and three-body force). Once the latter are put in normal order we obtain: (fn [H[f~n)= ( f n [ n + Heff[ fff), with Heft=-½
~ [F] 1/2 [Wm FI2 I'mr Cm
Z)
rr'r
[(AA)F(BB)r,]o W~W~' el
~(--)X[~k]l/2 x
{ F fr ] ~[B(BB)F']h] 0 F ' f X ) [[(AA)rA] "
(8)
Now we are entitled to speak of quasiconfigurations. All reference to the r-space has been eliminated and the problem reduces to a conventional diagonalization in the f-space, except for the existence of a three-body force. It is clear that eq. (8) can be interpreted as defining the interaction of quasiparticles f. Had we kept the form o f eq. (6) the choice of energy denominators would have been obvious (conceptually at least): h = single particle term + terms in Wffff, [4/frfr , Wrrrr .
(9)
It means that the blocks of the full matrix are taken to be diagonal. This choice cancels exactly C in eq. (5), thus insuring optimal convergence but it defines a different denominator for each intermediate state, does not allow the quasiconfiguration picture, and forces to calculate many matrix elements with standard shell-model techniques (including the cfp paraphernalia). To accomodate a cogent physical interpretation and good convergence, it is necessary to reach a compromise. We consider two possibilities: Approximation I takes as energy denominators the differences in single-particle energies and monopole isoscalar centroids between configurations fn and fn-m
-(m(n) = m [(ef - er) + (2n - m - 1)~/ff/2 - (n - m)Wfr - (m - 1)Wrr/2 , F . for the KB* interaction (defined after with Wi/= [ F ] - I 2; IF] (1 + 6ii) -1 Wi]i/,
(10) eq. (12)) we have the following
values: W f f = - 0 . 6 2 MeV,
WfP3/2 = - 0 " 1 7 M e V '
WfPl/2 = - 0 ' 2 2 MeV,
Wffs/2 = - 0 " 4 3 MeV.
These adopted values for the centroids are extensively discussed in ref. [4]. The term in Wrr is not well determined, but small enough in low order to be of no consequence. It is quite apparent that even for relatively small n the correction to the naive single-particle choice becomes large. Approximation H. Perfect convergence would occur with exact unperturbed energy denominators. We can call them gm(nJTX). As we have seen, the lifting of degeneracy through the extra quantum numbers X cannot be taken into account if we want to sum over intermediate states * 1. Approximation I amounts to assuming that the JT dependence of gis not very strong, and that the dispersion A for different X values at fixed JT is small (A/~-small). ,1 In fact, one can sum not only at fixed nJT, but also at fixed internal this possibility here.
JT couplings for the configurations. We shall not deal with 321
Volume 82B, number 3,4
PHYSICS LETTERS
9 April 1979
By and large, these conditions are fulfilled in the cases we have studied• However, it happens that for some J values the exact denominators are substantially smaller. As it does not involve too much extra numerical work we have made a choice that brings us quite close to the exact values:
gm(nJT) = exact values for fn-centroids at fixed JT for fn-m .
(11)
The prediagonalization in the fn space is not only feasible but advisable: the fn states never appear in intermediate sums, the dressing becomes clearer and some unwanted terms in higher order vanish. The explicit state dependence can be taken into account quite naturally. As for the fn-m configurations we are left with the Xdispersions only. The only trouble with the procedure is that no analytic expressions exist and the values of gm(nJT) have to be calculated numerically. To gauge the quality of the second-order results as compared with the (prohibitive) exact diagonalizations, we only notice that under the assumption of exact denominators the third-order matrix elements become simply: - n H I-~n (f~,l ¢~)3rd=~(ele2) -
-
-1
(f~lWllf~n-1 )(f,~n-1 IWllf~-2)(f~-2lW2lf~). n
(12)
Their effect is likely to be quite small and will be examined in a more complete paper. A full study of fn nuclei has been carried out, including calculations of transitions involving two-body effective operators. The Kuo-Brown [5] interaction was used, introducing only the modifications in monopole isoscalar and isovector centroids discussed in ref. [4]. We call it KB* (it differs from KB" in ref. [4] in that no other multipole changes were made in the f2 spectrum). Except for the statement that results are satisfactory (thus lending moral support to this communication) no details will be given here but for some very special nuclei, for which closed analytic expressions can be derived. Instead of assuming a 4°Ca core, we can make a particle-hole transform (interchange A and B operators) on neutrons (or neutrons and protons) and work on a 48Ca (or 56Ni) vacuum. With pure two-body forces the resulting fn spectra for 46Ca, 50Ti and 54Co are (obviously) identical to that of 42Sc (or whatever set of two-particle energies we choose). It is only in 48Sc that a Racah transformation gives the famous result of Pandya. When threebody forces are present the p - h transform is nontrivial everywhere. In addition to the usual one- and two-body contribution, the effective term in eq. (7) yields after some algebra the following corrections (we give only the contribution from multipole (W1) terms to which we will have to add the bare and effective pairing energies to obtain the spectra):
{i f
54Co: A E ( F ) = 2 ~ [F1][W[I[2 r,rx gl[f]
4
~ ( - 1 ) f+r-r2 --[PlP2] W[1W[ 2 r,rl, r2 gl
f
P1 f r
[J1]IWJI[2 + Z 3 IWIJI2 r,Jx gl[f] r gl
48Sc: A E ( J ) =
E
r,Jleven
l
--
322
gl[f]
E
r,J1
[J1] IW(ll 2 gl[f]
gl
1 ~
[J1][ WJx]2
r,J1
[glJ2]W(iWJ2{ff~l}{ffJ2 ]
+~r,&,& +2
[J1]IW1]'I~2+
gl
f r
( - 1 ) &+&+x [JIJ2 x] r, J 1 even, J2,x
gl
(14)
(_1)J1+J2 [JIJ2 x] r'Jl'J2'X
eI
(13)
F2
46Ca : AE(J) = same as 54Co (index meaning only spin; J1J2 even); 50Ti • A E ( J ) = ~
,"
WJ11WJ2Iff~l} {fffr~2}[ffJ I •
{ff~l} ff
( f f ~kJ ;
(is)
(16)
f rJ J
ff J21
Volume 82B, number 3,4
PHYSICS LETTERS 46Ca
AE (MeV)
9 April 1979 4aSC
&E ( M e W
6+ -
-
6+
4+
-
-
4
-
-
6 +
-
-
4+
_
_
2 +
4+
1 + _4+
+
2+
_
_
_
_
2 +
- - 7 ÷
2+
_ _ 2 -
-
3
+
2+
~
3+
"l
-
-
l 2 +
7+
7#
C
-
-
0+
(-'K).62) 42SC ( f 6 )
-
-
0+
-
-
0+
(-5.28)
(-6.55)
KB~,- ( f 6 )
KB.~- ( ~'6 )
-
-
0+
(-6.54) EX P
Fig. 1. Spectrum of 46Ca. In parentheses the nuclear binding energies with respect to 4°Ca. See also text after eq. (12).
50Ti
AE (MeV)
_
2 ~
-
3 +
-
5 +
6+
5~ _
_
-
-
4 ~
4 ~
_ _ , ¢ -
6
+
_ _ 6
(-2t.75)
(-t2.35)
(-13.9t)
42SC ( f S )
KB*(fS)
KB~(~8)
+
_
_
5+ 6 +
(-t3.35) EXP
Fig. 2. Same as fig. 1 for 48Sc.
AE(MeVI
~Co 6+
6+
6+ -
-
6+
-
4+
_
_
44" _
_
-
6 ~
_
-
-
6+
6
_
_
_
÷
_
-
-
4+
_
_
3 +
3 +
5÷
2+
_
_
,4 +
4 +
2+ 2+
4 ~-
2+
_
_ _ . 2
_
6 +
4 +
M
_
4 +
~, 74"
~
5+ 3
__2+ -
-
-
-
t
+
0 +
7 +
- 0 0+ (-40.94) 42SC ( f t O )
- - 0 (~23.88) KB*-(ftO1
+
0+ (-2.6.78) K B * { ~'t0)
Fig. 3. Same as fig. 1 for S°Ti.
-
-
(-25.92) EXP
0+
0+ (-94.16) 42SC ( f 4 4 )
+
t +
2+
3÷
2 +-
-
-
-
(-58.3'1) KB.(f44)
7+
7*
0+ (-6'/.42) K B ~ ,~ t 4 )
(-57.66) EXP
Fig. 4. Same as fig, 1 for 54Co.
Figs. 1 to 4 compare the level schemes obtained by (i) two-body fn calculations using experimental 425c values; (ii) two-body fn calculations using KB*; (iii) same as (ii) with the addition o f the corrective terms from eqs. ( 1 3 ) - ( 1 6 ) as well as the pairing contribution; approximation II is used (approximation I was found to be acceptable, with some exceptions, such as a strong overestimate o f the energy denominator for the 1+ state in 48 Sc); (iv) experiment. 323
Volume 82B, number 3,4
PHYSICS LETTERS
9 April 1979
It is clear in comparing the second and third columns in the figures ((ii) and (iii)) that the corrections to the bare force are large. Now some comments on individual cases. 46Ca. In this nucleus the effects of intruder states (see (c) above) are still very strong, which explains the relatively low positions of the calculated 4 + and 6 + levels. (In fact it is the 0 + and 2 + that are underbound, but we will not touch here on the fascinating problem of binding energies.) 48Sc. Here intruders play no role. Agreement with experiment is good (understatement). 5°Ti. Here intruders play no role either. But then: why is it that the spectrum is so similar to that of 42Sc where they do play a role? The answer to this long-standing puzzle is in the calculations, except that still we do not understand whether it is an accident or some profound message. 54 Co. The interaction is again doing a good job. We apologize for the fact that these four examples are not really illustrating degenerate perturbation theory since the model space is quite nondegenerate for each JT. However, for other nuclei that require diagonalization the situation is completely similar as far as agreement with experiment goes (the calculations are harder though). It is our hope that this very compact presentation will sustain two claims: (1) Degenerate perturbation theory can be formulated in a transparent way that avoids some of the usual problems of the Bloch-Horowitz approach: nonhermiticity of the interaction and the need of introducing folded diagrams. (2) Realistic interactions should not be tampered with, unless very profound reasons exist (the monopole effects for example). Our four examples show some remaining discrepancies with experiment. However, in no case changes of more than a few hundred keV seem to be necessary to correct the realistic matrix elements.
References [1 ] [2] [3] [4]
B.R. Barrett and M.W. Kirson, Advances in nuclear physics, Vol. 6, eds. M. Baranger and E. Vogt (Plenum, New York, 1973). J.H. Van Vleck, Phys. Rev. 33 (1929) section 4, p. 467. J.D. McCullen, B.F. Bayman and L. Zamick, Phys. Rev. 134 (1964) B515. E. Pasquini, Ph.D. Thesis, Strasbourg (1976); E. Pasquini and A. Zuker, Proc. Topical Conf. on Physics of medium light nuclei (Florence, 1977) eds. P. Blasi and R.A. Ricci (Editrice Compositori, Bologna, 1978) p. 62. [5] T.T.S. Kuo and G.E. Brown, Nucl. Phys. A114 (1968) 241. [6] P. Vold et al., Phys. Lett. 72B (1978) 311. [7] J.B. French, Proc. Intern. School of Physics "Enrico Fermi", Course XXXVI, ed. C. Bloch (Academic Press, London, 1966).
324
PHYSICS LETTERS
9 April 1979
QUASICONFIGURATIONS: AN APPROACH TO EFFECTIVE FORCES A. POVES, E. PASQUINI 1 and A.P. ZUKER Laboratoire de Physique Nucldaire Th(orique, CRN, 67037 Strasbourg Cedex, France
Received 7 January 1979
Many-body effective operators appear naturally by dressing states through a perturbative unitary transformation. They have forms that differ from those obtained in the Bloch-Horowitz approach. The fn/2 problem is treated explicitly. Pandya's transforms are generalized.
The success of the shell model can be equated to the possibility of describing properties in the vicinity of the ground states by the interaction between (few) particles confined in a small number of active orbits defined by a (hopefully) self-consistent calculation. For a long time it has been recognized that couplings to levels outside these basic configurations are important, and they have been the subject of extensive studies [1]. It is the purpose o f this paper to show that, provided perturbative methods are valid, such couplings can be accounted for rigorously, by introducing dressed many-particle states (quasiconfigurations). The idea is based on a revival of degenerate perturbation theory as explained in most QM books following Van Vleck's article of 1929 [2] Although it is quite feasible to cast our equations in a general framework, we have decided to present the problem using a notation that allows to visualize clearly the particular case we have dealt with in detail. Consider the f7/2 orbital (region from 40Ca to 56Ni) as spanning a well-defined model space. The validity of this assumption is borne out by classical calculations [3], by experimental fact and by careful SM studies [4]. Before coming to a diagonalization, the fn configurations have to be "dressed" to account for the following phenomena: (a) Realistic forces are always sufficiently singular to impose a G-matrix calculation. Many such calculations exist and have a striking common feature: different potentials fitting the two-nucleon data, subjected to different Brueckner treatments lead to extremely similar G-matrix elements [4]. We adopt for our calculations the classic K u o - B r o w n values [5] and refer to the discussions in ref. [4] for justification of the (only !) important change in monopole behavior. (b) The fp space couples to all possible multipole excitations of the core. The coupling may well be mediated by giant resonances, but it exists whether the latter exist or not. It is our opinion (to be sustained elsewhere) that for effects other than monopole, the bubble diagrams as used in KB [5] are a good approximation. (c) The fp space also couples to intruder states such as the famous 4 p - 2 h excitations in 42Ca for which experimental evidence is overwhelming [6]. Mixing is strong (but local) at the beginning of the shell because of vanishing energy denominators and perturbative treatment breaks down. Later on (after A = 46) the intruders move up and the fairly weak off-diagonal couplings allow to ignore them. As a consequence it is strongly advised to stop bothering about some individual states in the neighbourhood of 4°Ca and consider their fp centroids (well defined because of the locality o f the mixing). 1 Eduardo Pasquini disappeared in Argentina over two years ago and we have had no news of him (and his wife) since. Our results owe so much to his work, that his appearence as cosigner is more than justified. 319
Volume 82B, number 3,4
PHYSICS LETTERS
9 April 1979
(d) Once the preceding steps are followed we have a good fp model space and it only remains to diagonalize matrices ranging up to 10 8-dimensional. To solve the problem we take up again the hints of refs. [3,4] : fn configurations are sufficiently dominant and some perturbative treatment seems warranted. Now, we shall propose a scheme, adapted to the case at hand but capable of full generality through obvious extensions. First we need to say a word on the notation. We shall use throughout second quantization in the coupled representation following French's article [7]. The hamiltonian is
H = ~ei[i]l/2(AiBi) 0 i
~ [I']I/2w~I[(AiA/)r(B~BI)rlO~q~tc I . ijkl,I" i<~j,R<~l
(1)
ijkl run over the orbits f7/2 (from now on f) and f5/2 P3/2 Pl/2 (from now on, generically r). [' = J T i f w e use isospin representation. I" ~-J if we use np representation. [r] = ( 2 J + 1)(2T + 1) (or ( 2 J + 1)). ( _ ) r = (_)J+T (or (--)J)" ~ij = (1 + 8ij)-1/2. A is the usual creation operator a +, B is the tensor constructed out of the annihilation operator a.
B( " "i+iz tz = I.--) ai_iz . The basis for some n-particle level of given spin and isospin P, is given by the set: If n PXn), If n-1 r [,Xn_l), [fn-2r2FXn_2) etc., where X i are the extra quantum members specifying the states. For simplicity, we shall denote the basis by fn, fff-1, f,~-2 etc. At the beginning the hamiltonian spans the whole space. We are interested in decoupling some model block (fn) from the rest of the full (fr) n matrix. The typical matrix element (fnlHI fn-m) is a linear combination of Wfffr (multipole, m = 1) or Wffrr (pairing, m = 2) terms. We say it is of order W (O(I4/)). Now we dress the states fn (and fn-m in general) by introducing the perturbative unitary transformation -nf~ =f~+n
~
#,rn=l,2
(gc~ - -(o)-l(fnlVlfff -m) • tff-m + normalization ,
counterterms to O(V2/g 2) ,
(2)
and similar expressions for -(n-re. The splitting, H = h + V, between unperturbed (h with eigenvalues g) and perturbation (V) terms will be examined later. The following results obtain:
(-(n-mj-(~-m') = 8o~Smm, + O(V3/~-3),
m, m ' = 0, 1 , 2 ,
(3)
(fnIHl-f~-m)= Cm [ ( f n l H l f ~ - m ) + ~=~*(-ge~--g~,)-l(fnlHlf~+m-3)(f~+m-31Hlf~Tm)] + O ( V 3 / g 2 ) ,
(4)
where
Cm = 1 - ~ ( g c ~ ' - - g ~ ' ) - l ( ( f n l H l f n ' ) - ( f ~ ' - m l H l f ~ - m ) ) , ~'/J'
rn = 1 , 2 ,
+ { v, m=~l [(ga -- g3,)-1 + (gt3 - - ' - l " f n ' g ' f n - m ' < f . ~ - m l g l f ~ ) + O ( V 3 / g 3 ) {~-nIglf~ -n > =
(5)
•
(6)
Eq. (3) insures that we have a good unitary transformation to O(V3/~-3). Eqs. (4) and (5) show that the offdiagonal matrix between the dressed states ~-n and ~-~-rn have been reduced to O(V3/g 2) provided the energy denominators ~-, which we have not defined yet, are chosen to cancel Cm of eq. (5). To understand how this crucial choice is to be made, we first need to work on eq. (6). As it stands, it is Van Vleck's result [2] with one interesting improvement: the unperturbed states fn are not assumed to be degenerate and the energy denominators for the off-diagonal terms are nicely symmetric, insuring the hermiticity of the effective hamiltonian. Now, we are going to take advantage of the appearance, in eq. (6), of complete sets of intermediate states that will allow us to use closure, provided the energy denominators are constant for each complete set. As pairing and multipole excitations contribute in different ways we are going to separate them explicitly. We introduce the notations: 320
Volume 82B, number 3,4 W1 = Wfffr,
PHYSICS LETTERS
eml=~((ea---(Tm)-l+(-(6---(q,m)-l),
W2 = Wffrr ,
9 April 1979 m=l,2.
(7)
Here we have made explicit the m dependence of the intermediate states, while omitting any reference in gm to other quantum numbers. To invoke closure em has to be a constant for each complete set. Now we can separate f and r spaces in eq. (6) using the standard result for the reduced matrix element of coupled spherical tensors and sum over intermediate states. The contribution from r space reduces to a constant. For the pairing term four fermion operators remain (two-body force), while the multipole term has six (two- and three-body force). Once the latter are put in normal order we obtain: (fn [H[f~n)= ( f n [ n + Heff[ fff), with Heft=-½
~ [F] 1/2 [Wm FI2 I'mr Cm
Z)
rr'r
[(AA)F(BB)r,]o W~W~' el
~(--)X[~k]l/2 x
{ F fr ] ~[B(BB)F']h] 0 F ' f X ) [[(AA)rA] "
(8)
Now we are entitled to speak of quasiconfigurations. All reference to the r-space has been eliminated and the problem reduces to a conventional diagonalization in the f-space, except for the existence of a three-body force. It is clear that eq. (8) can be interpreted as defining the interaction of quasiparticles f. Had we kept the form o f eq. (6) the choice of energy denominators would have been obvious (conceptually at least): h = single particle term + terms in Wffff, [4/frfr , Wrrrr .
(9)
It means that the blocks of the full matrix are taken to be diagonal. This choice cancels exactly C in eq. (5), thus insuring optimal convergence but it defines a different denominator for each intermediate state, does not allow the quasiconfiguration picture, and forces to calculate many matrix elements with standard shell-model techniques (including the cfp paraphernalia). To accomodate a cogent physical interpretation and good convergence, it is necessary to reach a compromise. We consider two possibilities: Approximation I takes as energy denominators the differences in single-particle energies and monopole isoscalar centroids between configurations fn and fn-m
-(m(n) = m [(ef - er) + (2n - m - 1)~/ff/2 - (n - m)Wfr - (m - 1)Wrr/2 , F . for the KB* interaction (defined after with Wi/= [ F ] - I 2; IF] (1 + 6ii) -1 Wi]i/,
(10) eq. (12)) we have the following
values: W f f = - 0 . 6 2 MeV,
WfP3/2 = - 0 " 1 7 M e V '
WfPl/2 = - 0 ' 2 2 MeV,
Wffs/2 = - 0 " 4 3 MeV.
These adopted values for the centroids are extensively discussed in ref. [4]. The term in Wrr is not well determined, but small enough in low order to be of no consequence. It is quite apparent that even for relatively small n the correction to the naive single-particle choice becomes large. Approximation H. Perfect convergence would occur with exact unperturbed energy denominators. We can call them gm(nJTX). As we have seen, the lifting of degeneracy through the extra quantum numbers X cannot be taken into account if we want to sum over intermediate states * 1. Approximation I amounts to assuming that the JT dependence of gis not very strong, and that the dispersion A for different X values at fixed JT is small (A/~-small). ,1 In fact, one can sum not only at fixed nJT, but also at fixed internal this possibility here.
JT couplings for the configurations. We shall not deal with 321
Volume 82B, number 3,4
PHYSICS LETTERS
9 April 1979
By and large, these conditions are fulfilled in the cases we have studied• However, it happens that for some J values the exact denominators are substantially smaller. As it does not involve too much extra numerical work we have made a choice that brings us quite close to the exact values:
gm(nJT) = exact values for fn-centroids at fixed JT for fn-m .
(11)
The prediagonalization in the fn space is not only feasible but advisable: the fn states never appear in intermediate sums, the dressing becomes clearer and some unwanted terms in higher order vanish. The explicit state dependence can be taken into account quite naturally. As for the fn-m configurations we are left with the Xdispersions only. The only trouble with the procedure is that no analytic expressions exist and the values of gm(nJT) have to be calculated numerically. To gauge the quality of the second-order results as compared with the (prohibitive) exact diagonalizations, we only notice that under the assumption of exact denominators the third-order matrix elements become simply: - n H I-~n (f~,l ¢~)3rd=~(ele2) -
-
-1
(f~lWllf~n-1 )(f,~n-1 IWllf~-2)(f~-2lW2lf~). n
(12)
Their effect is likely to be quite small and will be examined in a more complete paper. A full study of fn nuclei has been carried out, including calculations of transitions involving two-body effective operators. The Kuo-Brown [5] interaction was used, introducing only the modifications in monopole isoscalar and isovector centroids discussed in ref. [4]. We call it KB* (it differs from KB" in ref. [4] in that no other multipole changes were made in the f2 spectrum). Except for the statement that results are satisfactory (thus lending moral support to this communication) no details will be given here but for some very special nuclei, for which closed analytic expressions can be derived. Instead of assuming a 4°Ca core, we can make a particle-hole transform (interchange A and B operators) on neutrons (or neutrons and protons) and work on a 48Ca (or 56Ni) vacuum. With pure two-body forces the resulting fn spectra for 46Ca, 50Ti and 54Co are (obviously) identical to that of 42Sc (or whatever set of two-particle energies we choose). It is only in 48Sc that a Racah transformation gives the famous result of Pandya. When threebody forces are present the p - h transform is nontrivial everywhere. In addition to the usual one- and two-body contribution, the effective term in eq. (7) yields after some algebra the following corrections (we give only the contribution from multipole (W1) terms to which we will have to add the bare and effective pairing energies to obtain the spectra):
{i f
54Co: A E ( F ) = 2 ~ [F1][W[I[2 r,rx gl[f]
4
~ ( - 1 ) f+r-r2 --[PlP2] W[1W[ 2 r,rl, r2 gl
f
P1 f r
[J1]IWJI[2 + Z 3 IWIJI2 r,Jx gl[f] r gl
48Sc: A E ( J ) =
E
r,Jleven
l
--
322
gl[f]
E
r,J1
[J1] IW(ll 2 gl[f]
gl
1 ~
[J1][ WJx]2
r,J1
[glJ2]W(iWJ2{ff~l}{ffJ2 ]
+~r,&,& +2
[J1]IW1]'I~2+
gl
f r
( - 1 ) &+&+x [JIJ2 x] r, J 1 even, J2,x
gl
(14)
(_1)J1+J2 [JIJ2 x] r'Jl'J2'X
eI
(13)
F2
46Ca : AE(J) = same as 54Co (index meaning only spin; J1J2 even); 50Ti • A E ( J ) = ~
,"
WJ11WJ2Iff~l} {fffr~2}[ffJ I •
{ff~l} ff
( f f ~kJ ;
(is)
(16)
f rJ J
ff J21
Volume 82B, number 3,4
PHYSICS LETTERS 46Ca
AE (MeV)
9 April 1979 4aSC
&E ( M e W
6+ -
-
6+
4+
-
-
4
-
-
6 +
-
-
4+
_
_
2 +
4+
1 + _4+
+
2+
_
_
_
_
2 +
- - 7 ÷
2+
_ _ 2 -
-
3
+
2+
~
3+
"l
-
-
l 2 +
7+
7#
C
-
-
0+
(-'K).62) 42SC ( f 6 )
-
-
0+
-
-
0+
(-5.28)
(-6.55)
KB~,- ( f 6 )
KB.~- ( ~'6 )
-
-
0+
(-6.54) EX P
Fig. 1. Spectrum of 46Ca. In parentheses the nuclear binding energies with respect to 4°Ca. See also text after eq. (12).
50Ti
AE (MeV)
_
2 ~
-
3 +
-
5 +
6+
5~ _
_
-
-
4 ~
4 ~
_ _ , ¢ -
6
+
_ _ 6
(-2t.75)
(-t2.35)
(-13.9t)
42SC ( f S )
KB*(fS)
KB~(~8)
+
_
_
5+ 6 +
(-t3.35) EXP
Fig. 2. Same as fig. 1 for 48Sc.
AE(MeVI
~Co 6+
6+
6+ -
-
6+
-
4+
_
_
44" _
_
-
6 ~
_
-
-
6+
6
_
_
_
÷
_
-
-
4+
_
_
3 +
3 +
5÷
2+
_
_
,4 +
4 +
2+ 2+
4 ~-
2+
_
_ _ . 2
_
6 +
4 +
M
_
4 +
~, 74"
~
5+ 3
__2+ -
-
-
-
t
+
0 +
7 +
- 0 0+ (-40.94) 42SC ( f t O )
- - 0 (~23.88) KB*-(ftO1
+
0+ (-2.6.78) K B * { ~'t0)
Fig. 3. Same as fig. 1 for S°Ti.
-
-
(-25.92) EXP
0+
0+ (-94.16) 42SC ( f 4 4 )
+
t +
2+
3÷
2 +-
-
-
-
(-58.3'1) KB.(f44)
7+
7*
0+ (-6'/.42) K B ~ ,~ t 4 )
(-57.66) EXP
Fig. 4. Same as fig, 1 for 54Co.
Figs. 1 to 4 compare the level schemes obtained by (i) two-body fn calculations using experimental 425c values; (ii) two-body fn calculations using KB*; (iii) same as (ii) with the addition o f the corrective terms from eqs. ( 1 3 ) - ( 1 6 ) as well as the pairing contribution; approximation II is used (approximation I was found to be acceptable, with some exceptions, such as a strong overestimate o f the energy denominator for the 1+ state in 48 Sc); (iv) experiment. 323
Volume 82B, number 3,4
PHYSICS LETTERS
9 April 1979
It is clear in comparing the second and third columns in the figures ((ii) and (iii)) that the corrections to the bare force are large. Now some comments on individual cases. 46Ca. In this nucleus the effects of intruder states (see (c) above) are still very strong, which explains the relatively low positions of the calculated 4 + and 6 + levels. (In fact it is the 0 + and 2 + that are underbound, but we will not touch here on the fascinating problem of binding energies.) 48Sc. Here intruders play no role. Agreement with experiment is good (understatement). 5°Ti. Here intruders play no role either. But then: why is it that the spectrum is so similar to that of 42Sc where they do play a role? The answer to this long-standing puzzle is in the calculations, except that still we do not understand whether it is an accident or some profound message. 54 Co. The interaction is again doing a good job. We apologize for the fact that these four examples are not really illustrating degenerate perturbation theory since the model space is quite nondegenerate for each JT. However, for other nuclei that require diagonalization the situation is completely similar as far as agreement with experiment goes (the calculations are harder though). It is our hope that this very compact presentation will sustain two claims: (1) Degenerate perturbation theory can be formulated in a transparent way that avoids some of the usual problems of the Bloch-Horowitz approach: nonhermiticity of the interaction and the need of introducing folded diagrams. (2) Realistic interactions should not be tampered with, unless very profound reasons exist (the monopole effects for example). Our four examples show some remaining discrepancies with experiment. However, in no case changes of more than a few hundred keV seem to be necessary to correct the realistic matrix elements.
References [1 ] [2] [3] [4]
B.R. Barrett and M.W. Kirson, Advances in nuclear physics, Vol. 6, eds. M. Baranger and E. Vogt (Plenum, New York, 1973). J.H. Van Vleck, Phys. Rev. 33 (1929) section 4, p. 467. J.D. McCullen, B.F. Bayman and L. Zamick, Phys. Rev. 134 (1964) B515. E. Pasquini, Ph.D. Thesis, Strasbourg (1976); E. Pasquini and A. Zuker, Proc. Topical Conf. on Physics of medium light nuclei (Florence, 1977) eds. P. Blasi and R.A. Ricci (Editrice Compositori, Bologna, 1978) p. 62. [5] T.T.S. Kuo and G.E. Brown, Nucl. Phys. A114 (1968) 241. [6] P. Vold et al., Phys. Lett. 72B (1978) 311. [7] J.B. French, Proc. Intern. School of Physics "Enrico Fermi", Course XXXVI, ed. C. Bloch (Academic Press, London, 1966).
324