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Large scale nuclear structure calculations with VAMPIR
Progress in Particle and Nuclear Physics PERGAMON
Progress in Particle and Nuclear Physics 46 (2001) 145-154 http://www.elsevier.nl/locate/npe
Large Scale Nuclear Structure Calculations with VAMPIR K. W. SCHMID 1. I n t r o d u c t i o n
lnstitut f6r Theoretische Physik der Universiffit Tffbingen, A u f der Morgenstelle 14, D-72076 Tiibingen, FRG
Many nuclear structure problems require the use of single particle basis-systems, which are far too large to allow for a complete diagonalization of a suitably chosen effective many-nucleon Hamiltonian as it is done in the Shell-model Configuration-Mixing (SCM) approach [1]. One is therefore forced to truncate the SCM space to a numerically feasible number of configurations. It has been tried to truncate according to the unperturbed energies of the configurations. This prescription, however, yields unsatisfying convergence properties especially for those states in the nuclear spectrum, which are of a more collective nature [2]. More promising is the use of Monte Carlo methods for the truncation as it has been demonstrated on this workshop by Otsuka [3]. Alternatively one can try to extract the relevant degrees of freedom directly from the nuclear Hamiltonian via variational procedures.
In this way the selection of the configurations is left entirely to the dynamics, and
(as in Otsukas approach) the ambiguities of the traditional truncation schemes are avoided. Out of the various possibilities to explore this avenue [4-6], the VAMPIR (Variation After Mean-field Projection In Realistic model spaces) approach [7,8], its extension for the description of excited states, the EXCITED VAMPIR [6,9], and finally the inclusion of additional correlations via the EXCITED FED VAMPIR method [10] are the most elaborate ones. In the VAMPIR approach the energetically lowest ("yrast") state with a given symmetry (i.e. fixed number of protons and neutrons, definite parity and angular momentum) is approximated by a single symmetry-projected Hartree-Fock-Bogoliubov (HFB) vacuum. The underlying mean-field is determined by a variational calculation after the projection. This yields the optimal description of each yrast-state in a symmetry-projected independent quasi-particle picture. The EXCITED VAMPIR approach is the straightforward extension of this method for the excited states with the same symmetry. Here for the first excited state of the considered system simply a second symmetryprojected HFB vacuum being Gram-Schmidt-orthogonalized to the yrast-solution is taken as test wave function. The variation yields then the optimal description of the first excited state again by a single projected determinant. In the same way afterwards the higher excited states are constructed. Finally the residual interaction is diagonalized in between all these solutions. This procedure has the advantage that one can describe excited states with a structure completely different from that of the corresponding yrast-state. The EXCITED FED VAMPIR approach, finally, uses several symmetryprojected HFB vacua instead of only one for the description of each state. It determines each of the different underlying HFB transformations successively together with the configuration mixing via 0146-6410/01/$ - see front matter © 2001 Elsevier Science BV. All rights reserved PII: SO146-6410(01)00118-1
146
K. W. Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
a chain of variational calculations. In this way it is ensured that each further symmetry-projected determinant does not disturb the wave function more than the last one added previously. These methods have been applied [11] with good success, e.g., to the rather complex shape-coexistence phenomena in the A ,-~ 70 mass-region. However, for a long time we still had imposed time-reversal invariance and axial symmetry on the underlying HFB transformations (see Ref. [6] and references therein). Some years ago these last restrictions could be removed [12]. For the first time results of symmetryprojected HFB calculations on the basis of completely unrestricted quasi-particle determinants have been obtained. Comparison with the results of complete shell-model diagonalizations in an ls0d-basis has shown that the exact results can be reproduced extremely well even if for each state only a single symmetry-projected determinant, i:e., an essentially "free" theory is used. This holds even in the middle of the shell where the shell-model dimensions are largest and, furthermore, the agreement is of the same quality for doubly-even, doubly-odd and odd nuclei. In the present contribution we shall now report on calculations within a lp0f-basis [13] and, in addition, discuss some of the problems with Galilean invariance encountered if model spaces exceeding one major shell are admitted. 2. O u t l i n e of t h e T h e o r y The theory of the VAMPIR approach has been discussed in detail elsewhere [8,9,10] and will hence be scetched only briefly in the following. We define our model-space by a finite Mb-dimensional set of Fermion creation and annihilation operators Ic L it ' c t k'
""J
1-Mb and {ci,ck,
""}Mb for
spherical single nucleon states.
The corresponding
vacuum l0 > is defined by c~10 > - 0 for all i = 1, ..., Mb. General quasi-particle creators are then introduced by
at
Mb -
J_
+
B,°c,)
(1)
i=1
This expression and the one for the corresponding annihilators can be combined easily to a single matrix-equation
(:')
where the (2Mb x 2Mb)-matrix F has to be unitary in order to ensure Fermion character for the quasi-particles. Eqs.(2) define the famous HFB transformation [14]. It is the most general linear transformation conserving the anti-commutation relations, which can be constructed within the chosen finite single particle basis. The corresponding vacuum IF > can be represented as M b~
(~1
where the product runs over all ~ with aa]0 > different from zero. Since the transformation (2) sums over all the quantum numbers characterizing the single particle basis states (isospin-projection, orbital angular momentum, total angular momentum, the 3-projection of the latter, and the radial excitation), IF > is neither an eigenstate of the square of the total angular
K. IV..Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
147
momentum operator zf2 nor of its 3-component ifz. Furthermore particle number and charge conservation are violated and, in general, the vacuum (3) has no definite parity either• The only symmetry still conserved is the so-called "number-parity" , i.e. IF > contains either only components with even or with odd total nucleon numbers A. From the vacuum (3) one can construct configurations with the desired symmetry quantum numbers s = A T z I '~ using the operator [6] ~K
-
P(IM;K)O~(2T.)O,(A)p(r) 2 I + 1 r(4,r)
1
r2,~
1 f 2 ~ d~oexp{i~o(A - .~)}~(1 • ~1 Jo + ~rfl)
(4)
where l~I is the parity-, .4 the nucleon number-, ~ / a n d Z the neutron and proton number- , i~(fl) the usual rotation-operator, and D ~ K ( f l ) its representation in angular momentum eigenstates. Via the K ~ u a n t u m number the configuration obtained by acting with the operator (4) on the HFB vacuum (3) does still depend on the orientation of the intrinsic quantisation axis. This unphysical dependence is eliminated by taking the linear combinations +I ICp;sM > =_ ~_,
O~MKIF>
fK;,
(5)
/{=--I
as physical configurations. Even if only a single determinant is considered, the restoration of the rotational symmetry thus introduces additional configuration-mixing coefficients f, which together with the intrinsic degrees of freedom of the underlying HFB transformation will have to be determined by variation. Restricting oneself to test wave functions of the form (5) one obtains the VAMPIR approach. The extension to linear combinations of several configurations of this type is straightforward [10]. Details of the procedure to determine the underlying mean fields and the configuration mixing by chains of variational calculations are discussed in the Refs. [6,10] and will hence not be repeated here. 3. U n r e s t r i c t e d versus S y m m e t r y l ~ s t r i c t e d M e t h o d s In a given basis the unrestricted, complex transformation F mixes all states regardless of their angular momentum quantum numbers, parity and proton or neutron origin. Thus after projection of parity, nucleon numbers, and finally the 3-dimensional projection of the total spin any type of state can be described in doubly-even, doubly-odd and odd nuclei already via a single determinant. This is not the case in the older versions of the approach where certain symmetries were imposed on the underlying HFB transformations. The requirement of axially symmetric HFB transformations induces that the vacua are eigenstates to the 3-component of the total angular momentum operator _1, with eigenvalues K = 0. The assumption of time-reversal invariance introduces in addition a two-fold degeneracy into the system. Consequently, the resulting test wave functions are restricted to even nucleon number and can only
148
K. W. Schmid IProg. Part. Nucl. Phys. 46 (2001) 145-154
describe states of doubly-even or doubly-odd nuclei. Furthermore, not even all states in these nuclei are accessible. Though by the use of essentially complex transformations all possible two-nucleon couplings are included, particular four- and more-nucleon couplings are missing [5,7,8] : two natural (or unnatural) parity pairs cannot be coupled to an unnatural parity four nucleon wave function and one natural and one unnatural parity pair not to a natural parity four nucleon state. Consequently, excitations which are dominated by configurations containing such "missing couplings" as irreducible substructures cannot be described even within the COMPLEX VAMPIR (CV) approach. In the earlier calculations we had imposed even more severe symmetry restrictions on the HFB transformations : proton-neutron- and parity-mixing were forbidden and only real mean-fields were admitted.
Consequently, only natural parity states in doubly~even nuclei were accessible by the
various so called REAL VAMPIR (RV) approaches. It should be stressed, however, that these deficiencies can be overcome even on the basis of symmetryrestricted transformations. This is done in the MONSTER approach, a multi-configuration method, which diagonalizes the Hamiltonian in the space of the VAMPIR solution and all the corresponding symmetry-projected two-quasi-particle excitations. In this way, K-mixing is included right from the beginning and missing couplings are avoided. The MONSTER approach, however, is only suited for exited states whose structure is not too different from that of the underlying HFB vacuum. 4. R e s u l t s
and Discussions
As first example for the quality of the unrestricted VAMPIR approach which we shall denote as GENERAL COMPLEX VAMPIR (GCV) in the following we shall discuss here some results [13] obtained using a lp0f-basis and the FPD6-force [15].
-260~
yrast-spectrum
',,
-261~ q
:------ 8"
/
::/ "';
- -
~.._____:
~ -264o
4"
.3"
-265 7_o
i
:i
(,,--~,,~:~
'
" 2"
",
>-
62 -26s- 30ZB32
2"
-269
!
io'i
RV
RM
CV
CM Method
GCV
i SCM(T2) SCM(TI)
While at the beginning of this shell the SCM dimensions are comparable with those in the middle of the ls0d-shell, it is no surprise that there the GCV approach performs excellently. So, e.g., for 46Ti energy differences of less than 100 KeV with respect to the exact results were obtained for all the
K. W. Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
149
considered states. More challenging are nuclei out of the middle of the shell where the SCM dimensions increase to the order 107 or even 10 s. Fig.1 displays the total binding energies relative to the 4°Ca core of the yrast spectrum of °~Zn obtained by 5 different approximate methods and compares them to two conventionally truncated SCM results presented in the last two columns. Starting from the left we first give the results of REAL VAMPIR (RV) calculations. Odd spins are not accessible in this approach as discussed in section 3. In the next column come the results of the REAL MONSTER (RM) which diagonalizes the chosen Hamiltonian in the space of the RV-vacuum obtained for the 0 + ground state and all corresponding symmetry-projected two-quasi-particle excitations. By construction therefore the total energy of the 0 + ground state remains unchanged while the higher spin states get some, though small, contributions from the two-quasi-particle excitations. Furthermore, in this case the symmetry-restrictions of the RV calculation are overcome and the odd spin states can be obtained as well. The third column from the left displays the results of the COMPLEX VAMP1R (CV) approach. Here odd spin states can be obtained even from the K = 0 vacuum, however, the figure clearly indicates that those states are dominated by four- and more-nucleon couplings which are "missing" in the still time-reversal invariant and axially symmetric vacuum. Thus their description is rather bad. For the even spin states the energy differences are of about the same magnitude as in the RV calculation, however, the absolute energy is considerably improved with respect to the latter approach. Again the shortcomings for the description of the odd spin states can be overcome by the corresponding multi-determinant COMPLEX MONSTER (CM) approach. In the next column we show the results of the (one-determinant) GCV calculations. Here drastic improvements for the absolute energies are observed. The last two colums show the results of two truncated SCM calculations (T1 allowed for at most two, T1 for at most one hole in the 0f7/2 shell). As can be seen, this severe trucation is rather bad. Already the RV approach produces more binding for the states up to spin 4 t h a n the SCM(T2) although the latter approach takes up to about 13000 configurations into account. The GCV ground state is even 4.5 MeV more bound than the truncated SCM(T2) one. -203.0
s~Ni ff ground state]
> fl)
O3
.>
-203.1-
>,
O3 ¢.ud
-203.2TSCM
QMC
GCV
FGCV(1) FGCV(2)
SCM
Obviously, the shell-model has been drastically truncated here (13000 configurations for a given spin state were about the maximum which could be treated with the OXBASH code on the best work
150
K. W. Schmid /Prog. Part. Nucl. Phys. 46 (2001) 145-154
station in Jyv~kyl~i). Modern implementations of the shell-model can do much better (up to about 25 Million configurations). This is documented for the 0 + ground state of 56Ni in Fig.2. From left to right the results of a truncated shell-model calculation (TSCM) allowing at most 6 holes in the 0t7/2 shell [16], the Quantum Monte Carlo (QMC) result of Otsuka [17], the GCV result, the results of the corresponding FED VAMPIR approach with one (FGCV(1)) and two (FGCV(2)) additional correlating determinants and finally the exact results (SCM) [18] are displayed. Note, that the vertical scale covers only 200 KeV so that the differences between the various methods are drastically enlarged. Already the one-determinant GCV result is only 159 KeV above the exact result. This is by no means trivial since for the exact result a few Million Configurations had to be taken into account, while in the GCV approach there are here only 1560 variational degrees of freedom. The difference from the exact energy decreases to only 76 KeV if altogether 3 projected determinant are used (FGCV(2)) and this latter solution is already 19 KeV lower than the QMC energy. One can conclude the GCV approach and its extensions provides even in the middle of the lp0f shell an excellent approximation to the exact SCM solutions and can very well compete here with the QMC approach. Compared to the shell-model approaches, however, the approaches of the VAMPIR family can be extended considerably easier to even much larger model spaces. 5. G a l i l e a n i n v a r i a n c e
As soon as the basis becomes larger than a single major shell, however, we encounter a tedious problem. Like most approaches to the nuclear many body problem we expand the wave functions in terms of (generalized) Slater determinants. In tMs way the Panli principle is fulfilled by construction but the Galilean invariance is severely broken. Its exact restoration for general bound states requires a projection into the center of momentum (COM) rest frame via the operator [19]
0(0) - f d3aexp{i~./5}
(..)
where/5 is the operator for the total linear momentum of the system. Obviously, as for the restoration (4) of the other symmetries, also this projection has to be done before the variation. For scattering states in addition the recoil of the continuum particle on the residual nucleus has to be treated [20]. It is usually argued that the restoration of Galilean invariance yields only an 1/A effect and is thus of minor importance at least for systems heavier than x60. We shall demonstrate with some examples that this statement is not true. For the sake of simplicity we shall do this for pure oscillator configurations. The first example are the spectroscopic factors for one-hole states in doubly-closed oscillator configurations. Here one expects spectroscopic factors of unity for all the possible hole-configurations. Projecting the A-nucleon ground state as well as the one-hole states with respect to it into their COM rest frames and taking furthermore the recoil of the continuum nucleon exactly into account we obtain considerable deviations from this simple picture. The results for the 3 nuclei aHe, 160 and a°Ca are displayed in Fig. 3. As can be seen, the lower shells are always depleted while the last shell below the Fermi level becomes "overoccupied in order that the total sum rule remains fulfilled.
151
K. W..Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
This overoccupation is a direct consequence of the fact the Galilean invariant description uses relative wave functions and can be understood easily. 1.1
o
1.0 ¸
.(2_ (3. o t3 t/} o
/ 0.9
G)
-" . . . . . . . . .........
(/) _e o 1-
......
"/--- ............... ,7' . . . . . . . . . . . . . . . . . . . _ _/Z..__.__ . . . . . . . . . . . . . . .
~--&-- Helium 4 "~ | + O x y g e n 16 ! l--I-
C a l c i u m 4 0 Jl.
,(____
0.8-
I
l
I
0
1 m a j o r shell n u m b e r N
2
As second example we consider the inelastic electron scattering from the 0d3/2 groundstate of 39K to the 0pl/2 excited state in the same nucleus. The corresponding Coulomb formfactors in the Galilean invariant and the normal description (with the latter one obviously including the usual approximate treatment of the COM motion via the Tassie-Barker factor [21]) are displayed in Fig. 4 (logarithmic scale on the left, linear scale on the right part of the figure). Again we see that the correct treatment of Galilean invariance is extremely important even for this rather simple example. •0.28
10"'~ "0.24 3 IO=-
=E
-0.20 10"3. r~
.o
E
-0.16
oE
-0.12
O ~e
-0.08
g
0
o= 10"o ~= 10~.
10",
- 0.04
I .--~- d3/2-p1/2 (normal) "~ d3/2 -pl/2 (projected)J~ 10.7
-0.08
0
1
2
3-momentum-transfer q / Fm'l
3
1
2
3-momentum-transfer q / Fm'l
Next we consider the energies of the one-hole states. As example we present in Fig. 5 the single particle energies in isO obtained in the normal (naturally including th subtraction of the COM
152
K. IV..Schmid /Prog. Part. Nucl. Phys. 46 (2001) 145-154
Hamiltonian) as well as in the Galilean invariaut prescription using the simple Brink-Boeker force (B1) [22] complemented with a short-range spin-orbit interaction derived from the Gogny D I S interaction [23].
tln•l:-
energies]
-10' 0pl/2
> Q~
-
-
-
-
,,Brink Boeker
BI i
•
-20. 0p3/2 c Q m
"6
-30.
1= r~
_e
-4o 0sl/2 - -50.
n - no
n - pr
p - no
p - pr
As expected, the (non-spurious) hole-states within the last occupied major shell are not affected, while considerable differences (about 6.5 MeV) are obtained for the l t ~ - h o l e states. Note, that both spectra, the normal one with spectroscopic factors of unity, the projected one with the spectroscopic factors out of Fig. 3 fulfill the so-called Kolthun-sum-rule. ~ o l e - energies~ (Gogny D1S~ -10-
I;ooo
-20-
lsl/2
]
0d3/2 - -
0d5/2 - -
(D ¢¢D
-30-6
0pl/2
f3. a) -40-
0p3/2
-
-
0sl/2
-
-
e-
-50-
-60
n - no
n - pr(rs)
p - no
p - pr(rs)
The situation becomes even more complicated, if density-dependent interactions are to be admitted. In this case the momentum transfer connected with the dependence of the interaction on the COM coordinate of two nucleons onto the remaining A-2 nucleons has to be taken into account.
K. w. Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
153
Obviously, for phenomenologieal interactions one may readjust the parameters in order to get the same contribution from the density-dependent term as normally. Unfortunately, this readjustment is strongly mass number and even state dependent and even if we tolerate that, because of the rather strong non-density dependent parts, the effects on the single-particle spectra are still larger than in the density-independent case. This can be seen from Fig. 6, were the one-hole spectrum for 4°Ca obtained with the Gogny D1S force in the normal and in the Galilean invariant description (rescaling the strength of the density dependent term as mentioned above) are compared. Still deviations of the order of 4 MeV are seen for both the 0p- as well as the 0s-holes. In the light of these results we have no doubt that in model spaces exceeding one major shell the restoration of full Galilean invariance in the various approaches of the VAMPIR family is unavoidable. 6. C o n c l u s i o n s a n d O u t l o o k We have reported the results of symmetry-projected HFB calculations in the lp0f-shell using entirely unrestricted vacua [13]. Comparison with the results of complete shell-model diagonalizations has shown that the exact results can be reproduced extremely well even if for each state only a single symmetry-projected determinant, i.e., an essentially "free" theory is used. This holds even in the middle of the shell where the shell-model dimensions are largest. Compared to the shell-model, however, the unrestricted GENERAL COMPLEX VAMPIR (GCV) can be easier extended to much larger basis systems. As in the shell-model, however, also in the GCV approach for such an extension a correct treatment of Galilean invariance is required. We have demonstrated by various examples that the effects of such a correct treatment are large and connot be neglected. Unfortunately, performing the projection into the COM rest frame before the variation leads to 3 (for density dependent interactions even 6) additional integrations and hence is hardly possible on present day sequential or vector computers. For parallel processing the situation is quite different. We have demonstrated in [12] that the multi-fold integrations to be performed in the GCV approach are particularly suited for parallel data processing. Since we have already succeeded in developing the mathematical apparatus needed for the projection of general HFB determinants into the center of momentum rest frame, we are confident that this procedure will become numerically feasible in the, though not in the very near, future. Acknowledgement The work reported here has been done in collaboration with Tuomas Hjelt, Esko Hammar~n and Amand Faessler. 7. R e f e r e n c e s [1] see, e.g., J.B.McGrory and B.H.Wildenthal, Annu.Rev.Nucl.Part.Sci. 30(1980)383 and references therein. [2] P.W.M.Glaudemans, in "Nuclear Structure at High Spin, Excitation, and Momentum Transfer", (H.Naun, Ed.), New York : Am.Inst.Phys., 1985, p.316 . [3] T.Otsuka, contribution to the Erice2000 workshop, in this volume
154
K. W. Schmid /Prog. Part. Nucl. Phys. 46 (2001) 145-154
[4] K.W.Schmid and F.Griimmer, Rep.Progr.Phys. 50(1987)731 . [5] K.W.Schmid, F.Griimmer, and A.Faessler, Ann.Phys.(NY) 180(1987)1. [6] K.W.Schmid in "Nuclear Structure Models", R.Beng"csson, J.Drayer, W.Nazaxewicz eds., World Scientific, Singapore, 1992, p. 333. [7] K.W.Schmid, F.Griimmer and A.Faessler, NucLPhys. A431(1984)205. [8] Zheng Ren-Rong, K.W.Sehmid, F.Griimmer and A.Faessler, Nucl.Phys. A494(1989)214. [9] K.W.Schmid, F.Griimmer, M.Kyotoku and A.Faessler, Nucl.Phys. A452(1986)493. [10] K.W.Schmid, Zheng Ren-Rong, F.Griimmer and A.Faessler, Nucl.Phys. A499(1989)63. [11] see, e.g., A.Petrovici, K.W.Schmid, A.Faessler, J.H.Hamilton and A.V.Ramayya Prog.Part.NucLPhys. 43(1999)485. [12] E.Hammax6n, K.W.Schmid and A.Faessler, Eur.Phys.J. A2(1998)371 . [13] T.Hjelt, K.W.Schmid, E.Hammax6n and A.Faessler, Eur.Phys.J. A7(2000)201 . [14] N.N.Bogoliubov, Zk.Eksp.Teor.Fiz. 34(1958)58 ; N.N.Bogoliubov, Usp.Fiz.Nauk. 67(1959)541 ; N.N.Bogoliubov and V.G.Soloviev, Dokl.Akaxi.Nauk. 124(1959)1011 . [15] W.A.Pdchter, M.G.van der Merve, R.E.Julies and B.A.Brown, Nucl.Phys. A523(1991)325. [16] cited in [17] as private communication with Caurier and Pores. [17] T.Otsuka, M.Honma and T.Mizusaki, Phys.Rev.Lett. 81(1998)1588. [18] A. Poves, private communication. [19] K.W.Schmid and F.Griiramer, Z.Phys. A336(1990)5 ; K.W.Schmid and P.-G.Reinhard, Nucl.Phys. A530(1991)283. [20] [21] [22] [23]
K.W.Schmid and G.Schmidt, Prog.Part.Nucl.Phys. 34(1995)361. L.J.Tassie and C.F.Baxker, Phys.Rev. 111(1958)940. D.M.Brink and E.Boeker, Nucl.Phys. 91(1967)1. J.F.Berger, M.Girod and D.Gogny, Comp.Phys.Comm. 63(1991)365 .
Progress in Particle and Nuclear Physics 46 (2001) 145-154 http://www.elsevier.nl/locate/npe
Large Scale Nuclear Structure Calculations with VAMPIR K. W. SCHMID 1. I n t r o d u c t i o n
lnstitut f6r Theoretische Physik der Universiffit Tffbingen, A u f der Morgenstelle 14, D-72076 Tiibingen, FRG
Many nuclear structure problems require the use of single particle basis-systems, which are far too large to allow for a complete diagonalization of a suitably chosen effective many-nucleon Hamiltonian as it is done in the Shell-model Configuration-Mixing (SCM) approach [1]. One is therefore forced to truncate the SCM space to a numerically feasible number of configurations. It has been tried to truncate according to the unperturbed energies of the configurations. This prescription, however, yields unsatisfying convergence properties especially for those states in the nuclear spectrum, which are of a more collective nature [2]. More promising is the use of Monte Carlo methods for the truncation as it has been demonstrated on this workshop by Otsuka [3]. Alternatively one can try to extract the relevant degrees of freedom directly from the nuclear Hamiltonian via variational procedures.
In this way the selection of the configurations is left entirely to the dynamics, and
(as in Otsukas approach) the ambiguities of the traditional truncation schemes are avoided. Out of the various possibilities to explore this avenue [4-6], the VAMPIR (Variation After Mean-field Projection In Realistic model spaces) approach [7,8], its extension for the description of excited states, the EXCITED VAMPIR [6,9], and finally the inclusion of additional correlations via the EXCITED FED VAMPIR method [10] are the most elaborate ones. In the VAMPIR approach the energetically lowest ("yrast") state with a given symmetry (i.e. fixed number of protons and neutrons, definite parity and angular momentum) is approximated by a single symmetry-projected Hartree-Fock-Bogoliubov (HFB) vacuum. The underlying mean-field is determined by a variational calculation after the projection. This yields the optimal description of each yrast-state in a symmetry-projected independent quasi-particle picture. The EXCITED VAMPIR approach is the straightforward extension of this method for the excited states with the same symmetry. Here for the first excited state of the considered system simply a second symmetryprojected HFB vacuum being Gram-Schmidt-orthogonalized to the yrast-solution is taken as test wave function. The variation yields then the optimal description of the first excited state again by a single projected determinant. In the same way afterwards the higher excited states are constructed. Finally the residual interaction is diagonalized in between all these solutions. This procedure has the advantage that one can describe excited states with a structure completely different from that of the corresponding yrast-state. The EXCITED FED VAMPIR approach, finally, uses several symmetryprojected HFB vacua instead of only one for the description of each state. It determines each of the different underlying HFB transformations successively together with the configuration mixing via 0146-6410/01/$ - see front matter © 2001 Elsevier Science BV. All rights reserved PII: SO146-6410(01)00118-1
146
K. W. Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
a chain of variational calculations. In this way it is ensured that each further symmetry-projected determinant does not disturb the wave function more than the last one added previously. These methods have been applied [11] with good success, e.g., to the rather complex shape-coexistence phenomena in the A ,-~ 70 mass-region. However, for a long time we still had imposed time-reversal invariance and axial symmetry on the underlying HFB transformations (see Ref. [6] and references therein). Some years ago these last restrictions could be removed [12]. For the first time results of symmetryprojected HFB calculations on the basis of completely unrestricted quasi-particle determinants have been obtained. Comparison with the results of complete shell-model diagonalizations in an ls0d-basis has shown that the exact results can be reproduced extremely well even if for each state only a single symmetry-projected determinant, i:e., an essentially "free" theory is used. This holds even in the middle of the shell where the shell-model dimensions are largest and, furthermore, the agreement is of the same quality for doubly-even, doubly-odd and odd nuclei. In the present contribution we shall now report on calculations within a lp0f-basis [13] and, in addition, discuss some of the problems with Galilean invariance encountered if model spaces exceeding one major shell are admitted. 2. O u t l i n e of t h e T h e o r y The theory of the VAMPIR approach has been discussed in detail elsewhere [8,9,10] and will hence be scetched only briefly in the following. We define our model-space by a finite Mb-dimensional set of Fermion creation and annihilation operators Ic L it ' c t k'
""J
1-Mb and {ci,ck,
""}Mb for
spherical single nucleon states.
The corresponding
vacuum l0 > is defined by c~10 > - 0 for all i = 1, ..., Mb. General quasi-particle creators are then introduced by
at
Mb -
J_
+
B,°c,)
(1)
i=1
This expression and the one for the corresponding annihilators can be combined easily to a single matrix-equation
(:')
where the (2Mb x 2Mb)-matrix F has to be unitary in order to ensure Fermion character for the quasi-particles. Eqs.(2) define the famous HFB transformation [14]. It is the most general linear transformation conserving the anti-commutation relations, which can be constructed within the chosen finite single particle basis. The corresponding vacuum IF > can be represented as M b~
(~1
where the product runs over all ~ with aa]0 > different from zero. Since the transformation (2) sums over all the quantum numbers characterizing the single particle basis states (isospin-projection, orbital angular momentum, total angular momentum, the 3-projection of the latter, and the radial excitation), IF > is neither an eigenstate of the square of the total angular
K. IV..Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
147
momentum operator zf2 nor of its 3-component ifz. Furthermore particle number and charge conservation are violated and, in general, the vacuum (3) has no definite parity either• The only symmetry still conserved is the so-called "number-parity" , i.e. IF > contains either only components with even or with odd total nucleon numbers A. From the vacuum (3) one can construct configurations with the desired symmetry quantum numbers s = A T z I '~ using the operator [6] ~K
-
P(IM;K)O~(2T.)O,(A)p(r) 2 I + 1 r(4,r)
1
r2,~
1 f 2 ~ d~oexp{i~o(A - .~)}~(1 • ~1 Jo + ~rfl)
(4)
where l~I is the parity-, .4 the nucleon number-, ~ / a n d Z the neutron and proton number- , i~(fl) the usual rotation-operator, and D ~ K ( f l ) its representation in angular momentum eigenstates. Via the K ~ u a n t u m number the configuration obtained by acting with the operator (4) on the HFB vacuum (3) does still depend on the orientation of the intrinsic quantisation axis. This unphysical dependence is eliminated by taking the linear combinations +I ICp;sM > =_ ~_,
O~MKIF>
fK;,
(5)
/{=--I
as physical configurations. Even if only a single determinant is considered, the restoration of the rotational symmetry thus introduces additional configuration-mixing coefficients f, which together with the intrinsic degrees of freedom of the underlying HFB transformation will have to be determined by variation. Restricting oneself to test wave functions of the form (5) one obtains the VAMPIR approach. The extension to linear combinations of several configurations of this type is straightforward [10]. Details of the procedure to determine the underlying mean fields and the configuration mixing by chains of variational calculations are discussed in the Refs. [6,10] and will hence not be repeated here. 3. U n r e s t r i c t e d versus S y m m e t r y l ~ s t r i c t e d M e t h o d s In a given basis the unrestricted, complex transformation F mixes all states regardless of their angular momentum quantum numbers, parity and proton or neutron origin. Thus after projection of parity, nucleon numbers, and finally the 3-dimensional projection of the total spin any type of state can be described in doubly-even, doubly-odd and odd nuclei already via a single determinant. This is not the case in the older versions of the approach where certain symmetries were imposed on the underlying HFB transformations. The requirement of axially symmetric HFB transformations induces that the vacua are eigenstates to the 3-component of the total angular momentum operator _1, with eigenvalues K = 0. The assumption of time-reversal invariance introduces in addition a two-fold degeneracy into the system. Consequently, the resulting test wave functions are restricted to even nucleon number and can only
148
K. W. Schmid IProg. Part. Nucl. Phys. 46 (2001) 145-154
describe states of doubly-even or doubly-odd nuclei. Furthermore, not even all states in these nuclei are accessible. Though by the use of essentially complex transformations all possible two-nucleon couplings are included, particular four- and more-nucleon couplings are missing [5,7,8] : two natural (or unnatural) parity pairs cannot be coupled to an unnatural parity four nucleon wave function and one natural and one unnatural parity pair not to a natural parity four nucleon state. Consequently, excitations which are dominated by configurations containing such "missing couplings" as irreducible substructures cannot be described even within the COMPLEX VAMPIR (CV) approach. In the earlier calculations we had imposed even more severe symmetry restrictions on the HFB transformations : proton-neutron- and parity-mixing were forbidden and only real mean-fields were admitted.
Consequently, only natural parity states in doubly~even nuclei were accessible by the
various so called REAL VAMPIR (RV) approaches. It should be stressed, however, that these deficiencies can be overcome even on the basis of symmetryrestricted transformations. This is done in the MONSTER approach, a multi-configuration method, which diagonalizes the Hamiltonian in the space of the VAMPIR solution and all the corresponding symmetry-projected two-quasi-particle excitations. In this way, K-mixing is included right from the beginning and missing couplings are avoided. The MONSTER approach, however, is only suited for exited states whose structure is not too different from that of the underlying HFB vacuum. 4. R e s u l t s
and Discussions
As first example for the quality of the unrestricted VAMPIR approach which we shall denote as GENERAL COMPLEX VAMPIR (GCV) in the following we shall discuss here some results [13] obtained using a lp0f-basis and the FPD6-force [15].
-260~
yrast-spectrum
',,
-261~ q
:------ 8"
/
::/ "';
- -
~.._____:
~ -264o
4"
.3"
-265 7_o
i
:i
(,,--~,,~:~
'
" 2"
",
>-
62 -26s- 30ZB32
2"
-269
!
io'i
RV
RM
CV
CM Method
GCV
i SCM(T2) SCM(TI)
While at the beginning of this shell the SCM dimensions are comparable with those in the middle of the ls0d-shell, it is no surprise that there the GCV approach performs excellently. So, e.g., for 46Ti energy differences of less than 100 KeV with respect to the exact results were obtained for all the
K. W. Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
149
considered states. More challenging are nuclei out of the middle of the shell where the SCM dimensions increase to the order 107 or even 10 s. Fig.1 displays the total binding energies relative to the 4°Ca core of the yrast spectrum of °~Zn obtained by 5 different approximate methods and compares them to two conventionally truncated SCM results presented in the last two columns. Starting from the left we first give the results of REAL VAMPIR (RV) calculations. Odd spins are not accessible in this approach as discussed in section 3. In the next column come the results of the REAL MONSTER (RM) which diagonalizes the chosen Hamiltonian in the space of the RV-vacuum obtained for the 0 + ground state and all corresponding symmetry-projected two-quasi-particle excitations. By construction therefore the total energy of the 0 + ground state remains unchanged while the higher spin states get some, though small, contributions from the two-quasi-particle excitations. Furthermore, in this case the symmetry-restrictions of the RV calculation are overcome and the odd spin states can be obtained as well. The third column from the left displays the results of the COMPLEX VAMP1R (CV) approach. Here odd spin states can be obtained even from the K = 0 vacuum, however, the figure clearly indicates that those states are dominated by four- and more-nucleon couplings which are "missing" in the still time-reversal invariant and axially symmetric vacuum. Thus their description is rather bad. For the even spin states the energy differences are of about the same magnitude as in the RV calculation, however, the absolute energy is considerably improved with respect to the latter approach. Again the shortcomings for the description of the odd spin states can be overcome by the corresponding multi-determinant COMPLEX MONSTER (CM) approach. In the next column we show the results of the (one-determinant) GCV calculations. Here drastic improvements for the absolute energies are observed. The last two colums show the results of two truncated SCM calculations (T1 allowed for at most two, T1 for at most one hole in the 0f7/2 shell). As can be seen, this severe trucation is rather bad. Already the RV approach produces more binding for the states up to spin 4 t h a n the SCM(T2) although the latter approach takes up to about 13000 configurations into account. The GCV ground state is even 4.5 MeV more bound than the truncated SCM(T2) one. -203.0
s~Ni ff ground state]
> fl)
O3
.>
-203.1-
>,
O3 ¢.ud
-203.2TSCM
QMC
GCV
FGCV(1) FGCV(2)
SCM
Obviously, the shell-model has been drastically truncated here (13000 configurations for a given spin state were about the maximum which could be treated with the OXBASH code on the best work
150
K. W. Schmid /Prog. Part. Nucl. Phys. 46 (2001) 145-154
station in Jyv~kyl~i). Modern implementations of the shell-model can do much better (up to about 25 Million configurations). This is documented for the 0 + ground state of 56Ni in Fig.2. From left to right the results of a truncated shell-model calculation (TSCM) allowing at most 6 holes in the 0t7/2 shell [16], the Quantum Monte Carlo (QMC) result of Otsuka [17], the GCV result, the results of the corresponding FED VAMPIR approach with one (FGCV(1)) and two (FGCV(2)) additional correlating determinants and finally the exact results (SCM) [18] are displayed. Note, that the vertical scale covers only 200 KeV so that the differences between the various methods are drastically enlarged. Already the one-determinant GCV result is only 159 KeV above the exact result. This is by no means trivial since for the exact result a few Million Configurations had to be taken into account, while in the GCV approach there are here only 1560 variational degrees of freedom. The difference from the exact energy decreases to only 76 KeV if altogether 3 projected determinant are used (FGCV(2)) and this latter solution is already 19 KeV lower than the QMC energy. One can conclude the GCV approach and its extensions provides even in the middle of the lp0f shell an excellent approximation to the exact SCM solutions and can very well compete here with the QMC approach. Compared to the shell-model approaches, however, the approaches of the VAMPIR family can be extended considerably easier to even much larger model spaces. 5. G a l i l e a n i n v a r i a n c e
As soon as the basis becomes larger than a single major shell, however, we encounter a tedious problem. Like most approaches to the nuclear many body problem we expand the wave functions in terms of (generalized) Slater determinants. In tMs way the Panli principle is fulfilled by construction but the Galilean invariance is severely broken. Its exact restoration for general bound states requires a projection into the center of momentum (COM) rest frame via the operator [19]
0(0) - f d3aexp{i~./5}
(..)
where/5 is the operator for the total linear momentum of the system. Obviously, as for the restoration (4) of the other symmetries, also this projection has to be done before the variation. For scattering states in addition the recoil of the continuum particle on the residual nucleus has to be treated [20]. It is usually argued that the restoration of Galilean invariance yields only an 1/A effect and is thus of minor importance at least for systems heavier than x60. We shall demonstrate with some examples that this statement is not true. For the sake of simplicity we shall do this for pure oscillator configurations. The first example are the spectroscopic factors for one-hole states in doubly-closed oscillator configurations. Here one expects spectroscopic factors of unity for all the possible hole-configurations. Projecting the A-nucleon ground state as well as the one-hole states with respect to it into their COM rest frames and taking furthermore the recoil of the continuum nucleon exactly into account we obtain considerable deviations from this simple picture. The results for the 3 nuclei aHe, 160 and a°Ca are displayed in Fig. 3. As can be seen, the lower shells are always depleted while the last shell below the Fermi level becomes "overoccupied in order that the total sum rule remains fulfilled.
151
K. W..Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
This overoccupation is a direct consequence of the fact the Galilean invariant description uses relative wave functions and can be understood easily. 1.1
o
1.0 ¸
.(2_ (3. o t3 t/} o
/ 0.9
G)
-" . . . . . . . . .........
(/) _e o 1-
......
"/--- ............... ,7' . . . . . . . . . . . . . . . . . . . _ _/Z..__.__ . . . . . . . . . . . . . . .
~--&-- Helium 4 "~ | + O x y g e n 16 ! l--I-
C a l c i u m 4 0 Jl.
,(____
0.8-
I
l
I
0
1 m a j o r shell n u m b e r N
2
As second example we consider the inelastic electron scattering from the 0d3/2 groundstate of 39K to the 0pl/2 excited state in the same nucleus. The corresponding Coulomb formfactors in the Galilean invariant and the normal description (with the latter one obviously including the usual approximate treatment of the COM motion via the Tassie-Barker factor [21]) are displayed in Fig. 4 (logarithmic scale on the left, linear scale on the right part of the figure). Again we see that the correct treatment of Galilean invariance is extremely important even for this rather simple example. •0.28
10"'~ "0.24 3 IO=-
=E
-0.20 10"3. r~
.o
E
-0.16
oE
-0.12
O ~e
-0.08
g
0
o= 10"o ~= 10~.
10",
- 0.04
I .--~- d3/2-p1/2 (normal) "~ d3/2 -pl/2 (projected)J~ 10.7
-0.08
0
1
2
3-momentum-transfer q / Fm'l
3
1
2
3-momentum-transfer q / Fm'l
Next we consider the energies of the one-hole states. As example we present in Fig. 5 the single particle energies in isO obtained in the normal (naturally including th subtraction of the COM
152
K. IV..Schmid /Prog. Part. Nucl. Phys. 46 (2001) 145-154
Hamiltonian) as well as in the Galilean invariaut prescription using the simple Brink-Boeker force (B1) [22] complemented with a short-range spin-orbit interaction derived from the Gogny D I S interaction [23].
tln•l:-
energies]
-10' 0pl/2
> Q~
-
-
-
-
,,Brink Boeker
BI i
•
-20. 0p3/2 c Q m
"6
-30.
1= r~
_e
-4o 0sl/2 - -50.
n - no
n - pr
p - no
p - pr
As expected, the (non-spurious) hole-states within the last occupied major shell are not affected, while considerable differences (about 6.5 MeV) are obtained for the l t ~ - h o l e states. Note, that both spectra, the normal one with spectroscopic factors of unity, the projected one with the spectroscopic factors out of Fig. 3 fulfill the so-called Kolthun-sum-rule. ~ o l e - energies~ (Gogny D1S~ -10-
I;ooo
-20-
lsl/2
]
0d3/2 - -
0d5/2 - -
(D ¢¢D
-30-6
0pl/2
f3. a) -40-
0p3/2
-
-
0sl/2
-
-
e-
-50-
-60
n - no
n - pr(rs)
p - no
p - pr(rs)
The situation becomes even more complicated, if density-dependent interactions are to be admitted. In this case the momentum transfer connected with the dependence of the interaction on the COM coordinate of two nucleons onto the remaining A-2 nucleons has to be taken into account.
K. w. Schmid / Prog. Part. Nucl. Phys. 46 (2001) 145-154
153
Obviously, for phenomenologieal interactions one may readjust the parameters in order to get the same contribution from the density-dependent term as normally. Unfortunately, this readjustment is strongly mass number and even state dependent and even if we tolerate that, because of the rather strong non-density dependent parts, the effects on the single-particle spectra are still larger than in the density-independent case. This can be seen from Fig. 6, were the one-hole spectrum for 4°Ca obtained with the Gogny D1S force in the normal and in the Galilean invariant description (rescaling the strength of the density dependent term as mentioned above) are compared. Still deviations of the order of 4 MeV are seen for both the 0p- as well as the 0s-holes. In the light of these results we have no doubt that in model spaces exceeding one major shell the restoration of full Galilean invariance in the various approaches of the VAMPIR family is unavoidable. 6. C o n c l u s i o n s a n d O u t l o o k We have reported the results of symmetry-projected HFB calculations in the lp0f-shell using entirely unrestricted vacua [13]. Comparison with the results of complete shell-model diagonalizations has shown that the exact results can be reproduced extremely well even if for each state only a single symmetry-projected determinant, i.e., an essentially "free" theory is used. This holds even in the middle of the shell where the shell-model dimensions are largest. Compared to the shell-model, however, the unrestricted GENERAL COMPLEX VAMPIR (GCV) can be easier extended to much larger basis systems. As in the shell-model, however, also in the GCV approach for such an extension a correct treatment of Galilean invariance is required. We have demonstrated by various examples that the effects of such a correct treatment are large and connot be neglected. Unfortunately, performing the projection into the COM rest frame before the variation leads to 3 (for density dependent interactions even 6) additional integrations and hence is hardly possible on present day sequential or vector computers. For parallel processing the situation is quite different. We have demonstrated in [12] that the multi-fold integrations to be performed in the GCV approach are particularly suited for parallel data processing. Since we have already succeeded in developing the mathematical apparatus needed for the projection of general HFB determinants into the center of momentum rest frame, we are confident that this procedure will become numerically feasible in the, though not in the very near, future. Acknowledgement The work reported here has been done in collaboration with Tuomas Hjelt, Esko Hammar~n and Amand Faessler. 7. R e f e r e n c e s [1] see, e.g., J.B.McGrory and B.H.Wildenthal, Annu.Rev.Nucl.Part.Sci. 30(1980)383 and references therein. [2] P.W.M.Glaudemans, in "Nuclear Structure at High Spin, Excitation, and Momentum Transfer", (H.Naun, Ed.), New York : Am.Inst.Phys., 1985, p.316 . [3] T.Otsuka, contribution to the Erice2000 workshop, in this volume
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K. W. Schmid /Prog. Part. Nucl. Phys. 46 (2001) 145-154
[4] K.W.Schmid and F.Griimmer, Rep.Progr.Phys. 50(1987)731 . [5] K.W.Schmid, F.Griimmer, and A.Faessler, Ann.Phys.(NY) 180(1987)1. [6] K.W.Schmid in "Nuclear Structure Models", R.Beng"csson, J.Drayer, W.Nazaxewicz eds., World Scientific, Singapore, 1992, p. 333. [7] K.W.Schmid, F.Griimmer and A.Faessler, NucLPhys. A431(1984)205. [8] Zheng Ren-Rong, K.W.Sehmid, F.Griimmer and A.Faessler, Nucl.Phys. A494(1989)214. [9] K.W.Schmid, F.Griimmer, M.Kyotoku and A.Faessler, Nucl.Phys. A452(1986)493. [10] K.W.Schmid, Zheng Ren-Rong, F.Griimmer and A.Faessler, Nucl.Phys. A499(1989)63. [11] see, e.g., A.Petrovici, K.W.Schmid, A.Faessler, J.H.Hamilton and A.V.Ramayya Prog.Part.NucLPhys. 43(1999)485. [12] E.Hammax6n, K.W.Schmid and A.Faessler, Eur.Phys.J. A2(1998)371 . [13] T.Hjelt, K.W.Schmid, E.Hammax6n and A.Faessler, Eur.Phys.J. A7(2000)201 . [14] N.N.Bogoliubov, Zk.Eksp.Teor.Fiz. 34(1958)58 ; N.N.Bogoliubov, Usp.Fiz.Nauk. 67(1959)541 ; N.N.Bogoliubov and V.G.Soloviev, Dokl.Akaxi.Nauk. 124(1959)1011 . [15] W.A.Pdchter, M.G.van der Merve, R.E.Julies and B.A.Brown, Nucl.Phys. A523(1991)325. [16] cited in [17] as private communication with Caurier and Pores. [17] T.Otsuka, M.Honma and T.Mizusaki, Phys.Rev.Lett. 81(1998)1588. [18] A. Poves, private communication. [19] K.W.Schmid and F.Griiramer, Z.Phys. A336(1990)5 ; K.W.Schmid and P.-G.Reinhard, Nucl.Phys. A530(1991)283. [20] [21] [22] [23]
K.W.Schmid and G.Schmidt, Prog.Part.Nucl.Phys. 34(1995)361. L.J.Tassie and C.F.Baxker, Phys.Rev. 111(1958)940. D.M.Brink and E.Boeker, Nucl.Phys. 91(1967)1. J.F.Berger, M.Girod and D.Gogny, Comp.Phys.Comm. 63(1991)365 .