DSpace at VNU: The higher Tamm-Dancoff approximation: Theoretical context and phenomenological aspects

January 28, 2008 10:41 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only International Journal of Modern Physics E Vol 17, No (2008) 228–239 c World Scientific Publishing Company THE HIGHER TAMM–DANCOFF APPROXIMATION: THEORETICAL CONTEXT AND PHENOMENOLOGICAL ASPECTS PHILIPPE QUENTIN∗ Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA HOUDA NAIDJA Centre d’Etudes Nucl´ eaires de Bordeaux-Gradignan, Universit´ e Bordeaux-I and CNRS/IN2P3, BP 120, 33175 Gradignan, France LUDOVIC BONNEAU∗ Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA JOHANN BARTEL Institut Pluridisciplinaire Hubert Curien, CNRS/IN2P3 and Universit´ e Louis Pasteur, 23 Rue du Loess, BP 28 67037 Strasbourg, France HA THUY LONG Vietnam National University in Hanoi, College of Natural Sciences, 334 Nguyen Trai, Hanoi, Vietnam Received October 22, 2007 Revised November 22, 2007 We present the key aspects of the theoretical foundations of the Higher Tamm–Dancoff Approximation which can be interpreted as a truncated shell-model approach based on a Hartree–Fock solution, ensuring the conservation of the particle number Then we discuss some phenomenological aspects of the residual interactions used, namely the delta interaction to describe the neutron–neutron and proton–proton pairing correlations and the quadrupole–quadrupole interaction to describe vibrational correlations Introduction The Higher Tamm–Dancoff Approximation,1–7 hereafter referred to as HTDA, is a microscopic approach aiming at treating in a tractable and physically sound ∗ Permanent address: Centre d’Etudes Nucl´ eaires de Bordeaux-Gradignan, Universit´e Bordeaux-I and CNRS/IN2P3, BP 120, 33175 Gradignan, France 228 January 28, 2008 10:41 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only The Higher Tamm–Dancoff Approximation 229 fashion nucleonic correlations beyond mean field solutions In doing so one intends also to take stock of the phenomenological successes of effective nucleon-nucleon interactions of the Gogny or Skyrme type Being in essence a highly truncated shell model calculation, designed to treat simultaneously, at least in principle, correlations issuing from pairing, vibrations (RPA-like) and long range collective modes or those stemming from necessary symmetry restorations, the HTDA guarantees by construction the particle-number conservation On the other hand, the very success of this truncation scheme, i.e its fast convergence in the particle-hole basis, lies in its capacity to choose as appropriately as possible the underlying Slater determinant vacuum state This is actually performed by retaining for the latter a self-consistent Hartree–Fock solution resulting from an effective-force one-body reduction associated with a one-body density matrix which includes in a self-consistent (yet non-variational) fashion the one-body effects of nucleonic correlations In the next section we present the key aspects of the theoretical foundations of the HTDA approach and in the following sections we discuss phenomenological aspects of the residual interactions used to describe pairing correlations in the |Tz | = channel and vibrational correlations, namely a delta interaction for the former (Sect 3) and an isoscalar quadrupole–quadrupole interaction for the latter (Sect 4) In Sect we draw the main conclusions of this work Theoretical Context Let us now pay a special attention to the precise definition of the Hamiltonian H A priori, given an effective interaction v and any mean field V , the total Hamiltonian H = K + v (where K is the kinetic energy operator) may be trivially split into a shell-model part Hsm = K + V and a residual interaction part vres = v − V In practice however, things are not so crystal-clear for at least three reasons First of all, if one uses for v an interaction fitted so as to yield good Hartree–Fock mean fields there is absolutely no guarantee (and in most cases this caveat is fully justified) that it yields at the same time the correct p − p and h − h matrix elements needed to describe realistically the correlation properties A second difficulty arises from the fact that one often takes only into account a part of the full residual interaction (as when dealing only with the |Tz | = part of the pairing correlations, thus ignoring the effects of the low-multipole part of vres and the neutron-proton correlations altogether) Finally, in practice to keep the computational load within reasonable limits, one treats the correlations in a single-particle (sp) space which is by far more limited than the one used for building the mean field Specifically we shall use, in our case, a Skyrme effective nucleon-nucleon interaction and thus start from a Hamiltonian HSk = K + vSk , (1) where vSk stands for a Skyrme effective interaction Now, given a generic Slater determinant |Φ yielding the one-body reduction vSk of vSk , we consider here, as in January 28, 2008 10:41 230 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 P Quentin et al previous HTDA calculations, that HSk could be written as HSk = Hsm + vres , (2) with the following definitions of Hsm and vres : Hsm = K + vSk − Φ|vSk |Φ (3) vres = vSk − vSk + Φ|vSk |Φ (4) Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only and The introduction in the above of the expectation values of vSk for the state |Φ is necessitated by the requirement that Φ|HSk |Φ = Φ|Hsm |Φ (5) Φ|vres |Φ = (6) or equivalently Now, let us perform a multipole expansion of vres A traditional approximation (see, e.g., Ref 8) of the high-multipole part of the latter consists in representing it as a δ interaction This part of vres is known to generate mostly (but of course not exclusively) pairing correlations The remaining part of vres thus involves a sum of its lowest multipole components which are generally considered to yield the socalled RPA correlations This approximate scheme is the physical rationale behind, for instance, the pairing-plus-quadrupole model.9,10 In our case we will, in practice, follow this approach and approximate the “exact” expression of vres given in Eq (4) upon replacing in it vSk by a phenomenological interaction w defined schematically by λ=λmax µ=λ w = Aδ + χλ Qλ,µ Qλ,−µ , (7) λ=λmin µ=−λ where the delta force may be considered to be spin-dependent (i.e with A including spin operators) or equivalently (by virtue of the Pauli principle and of the space-even character of the δ interaction) isospin-dependent Furthermore the Q · Q part of w may be simplified upon assuming that the χλ terms, which may be isospin-dependent, are merely constant (with respect to the relative distance of the nucleons) We define now the HTDA Hamiltonian H, which differs from the original HSk due to the introduction of w (Eq (7)) as the 2-body interaction to be used in the definition of vres (Eq (4)) instead of vSk Similarly to what has been done previously,6 we may rewrite H as H = Φ|HSk |Φ + HIQP + vres , (8) where HIQP is a kind of independent quasi-particle Hamiltonian standing for Ei ηi† ηi HIQP = i (9) January 28, 2008 10:41 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only The Higher Tamm–Dancoff Approximation 231 In the above, the sum runs over the whole spectrum (S) of K + vSk The energy Ei is equal to ei (−ei resp.) if the state |i is a particle (hole resp.) state with respect to the vacuum |Φ Similarly the creation operator ηi† stands for the operator a†i (ai resp.) depending on the particle (hole resp.) character of the state |i The definition of H given by Eq (8), for practically useful and physically appealing as it appears to be, has the unwanted feature of being not consistently defined for its two parts (Hsm , vres ) and thus being state-dependent, namely depending on the vacuum |Φ Therefore we will have to specify which HTDA Hamiltonian we will have to take into account, namely considering H (ν) associated with the vacuum state |Φν : (ν) (ν) H (ν) = Φν |HSk |Φν + HIQP + vres , (10) where the dependence of HIQP and vres on |Φν stems straightforwardly from their definitions In this paper, we will concentrate on the phenomenological determination of the δ part of the residual interaction and of one particular component of its Q · Q part, namely its quadrupole part (thus related to λ = 2) Phenomenology of the Delta Residual Interaction In this paper we shall limit ourselves to the “ordinary” neutron-neutron (n-n) and proton-proton (p-p) pairing correlations, using a zero-range force The corresponding interaction w to be used in the definition of vres will therefore write as aδ (11) w = (1 − P σ ) δ(r1 − r2 ) Our goal here will thus be to present and discuss a practical procedure to determine the aδ so that the corresponding residual interaction allows a realistic description of these n-n and p-p correlations There are, as well known since the seminal papers of Kisslinger and Sorensen,10,11 various quantities which may be used for that purpose The most frequently used path to so consists in fitting the experimentally observed odd-even energy differences which we will discuss in some detail first before explaining what are the theoretical calculations to be performed accordingly 3.1 Experimentally Deduced Odd-Even Mass Differences Suppose that we are dealing with a number N of fermions which is furthermore, here as in the following, an even number Note that in the following we will consider explicitly one type of fermions only One may, for instance, consider that the development below is concerned with neutrons and therefore, at each time where a notation like N , N + appears, it stands for (N, Z), (N + 1, Z) in a shorthand notation The simplest quantity which may be considered is (0) (12) δ3 = E(N + 1) + E(N − 1) − E(N ) , January 28, 2008 10:41 Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only 232 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 P Quentin et al where E(N ) is the total energy of the N -fermion system The superscript (0) indicates that we are computing the odd-even difference precisely for N fermions while the subscript refers here to the fact that we are using, for that purpose, the masses of three nuclei Indeed, the above Eq (12) may be interpreted in the following “geometrical” way One assumes that the total energy of our fermionic system varies analytically with the number N of fermions, however according to two different functions (E (e) (N ) and (E (o) (N )): the former for even numbers N of fermions, the latter for odd numbers N of fermions which are furthermore deduced (0) from each other by a constant shift δ3 If one makes locally a linear approximation for these energy curves one readily obtains Eq (12) for the shift One could, however, compute similarly for the N + (N − resp.) systems the (+) (−) following differences δ3 (δ3 resp.) defined as (+) δ3 = E(N + 1) − E(N + 2) − E(N ) (13) = E(N − 1) − E(N ) − E(N − 2) (14) and (−) δ3 Obviously if the approximations made (linearity of E(N ) and energy curves corresponding to odd and even N being deduced by translation from one another) in the evaluation of the shift of Eq (12) were to hold perfectly, the three quantities (0) (+) (−) δ3 , δ3 and δ3 would be identical In practice while keeping the translation approximation one may account for a more complicated local approximation for E(N ) and suppose that they are in fact polynomials of order in N One would then obtain the so-called (see, e.g., Eq (21) of Ref 12) fourth-order finite difference shifts defined (with transparent notation) as (0) δ5 = − E(N + 2) + E(N − 2) + E(N + 1) + E(N − 1) − 6E(N ) (15) In the above evaluations, one makes stock on the contention that the shift between the odd and even layers of the energy surfaces is constant and that the binding energies are smooth functions of N and Z, at least locally Clearly, any rapid energy variations related to the shell structure might prevent these approximate considerations to hold This is why the five-point determination might be considered as better than the three-point one, in that it could be a priori more able to smooth (0) (0) out sudden variations One may also argue that δ5 and δ3 are quantities better suited to our purpose because they are centered by construction on the even-even nucleus under consideration However one has to take into account one particular aspect of our calculations here As a matter of fact we are studying even-even nuclei such that N = Z and their immediate neighbors It so happens that a systematic binding energy pattern is present there known as the Wigner term parametrized in, e.g., Ref 13 as: EWigner = −30 I MeV , (16) January 28, 2008 10:41 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only The Higher Tamm–Dancoff Approximation 233 where I is the relative neutron excess I = (N − Z)/A with usual notation This means that for each increase of half a unit in the absolute value of Tz one looses 30A−1 MeV, at least in the vicinity of Tz = 0, beyond what the smooth variation that the liquid drop or droplet would provide This Wigner term contribution to the binding energy has nothing to with the odd-even mass difference under scrutiny here It has thus to be removed It is clear from Eqs (12), (13), (14), and (15) that, (0) (0) for even N = Z values, while δ3 and δ5 are affected by this Wigner structural (+) (−) effect, δ3 and δ3 are not This is why, upon further demanding a symmetrized definition of the odd-even energy shift, we will attempt, in the following, to theoretically describe the particular experimental quantity (given in nuclear mass tables as in Ref 14) defined as δ= (+) (−) δ + δ3 (17) around even-even Tz = isotopes 3.2 Theoretical Evaluation of the Odd-Even Energy Shifts δ Within the BCS or HFB context, one is used to approximately consider δ (see Refs 10,11) as being equal to the lowest relevant quasi-particle energy or, even more approximately, to the pairing gap associated with that quasi-particle state For the sake of putting in proper perspective our HTDA method, we will briefly review how δ is evaluated within the BCS approximation One defines the odd nuclear system having N ± particles from a BCS solution for N fermions, |BCSN , as a one quasi-particle wave-function |BCSkN corresponding to the sp state |k as |BCSkN = ηk† |BCSN , (18) where ηk† is the operator which creates the quasi-particle state |k The expectation ˆ for |BCSkN is given by value N k of the particle number N N k = N + u2k − vk2 , (19) thus describing the N + (N − resp.) system whenever k is a particle (hole resp.) state with u2k ∼ (vk2 ∼ resp.) It is of interest for our further developments to note that |BCSkN = a†k |BCSN (k) , (20) where |BCSN (k) is a BCS-like state in the standard definition of which the operator (uk + vk a†k a†k¯ ) has been removed This state does not, therefore, contain in general exactly N particles The energy associated with the one quasi-particle state |BCS kN , relative to the energy of |BCSN is E(BCSkN ) − E(BCSN ) = λ(u2k − vk2 ) + Ek , (21) January 28, 2008 Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only 234 10:41 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 P Quentin et al where λ is the chemical potential and Ek the relevant quasi-particle energy Upon (±) assuming that E(BCSN ±2 ) = E(BCSN ) ± 2λ, one readily finds that δ3 = Ek , when describing approximately, as a one quasi-particle state on |BCSN , the N + or N − system depending on the value of vk2 In the HTDA case, now, we proceed in the following way We start with a Hartree–Fock solution for N fermions (N even) whose dynamics is governed by a Hamiltonian H, corresponding to a Slater determinant |ΦN Upon performing a standard HTDA calculation, one gets a correlated solution |ΨN In the following, we will denote by |a the last occupied state and by |α the first unoccupied state associated with the sp spectrum (S) corresponding to the Slater determinant |ΦN , whose sp energies are ea and eα , respectively The energy E(N ) appearing in Eq (12) is nothing but ΨN |H (N ) |ΨN , where H is defined in Eq (10) with ν = N The main effort described below is an attempt to produce reasonable approximations within the HTDA framework for the state |ΨN +1 (|ΨN −1 resp.) in which the expectation value of H is E(N + 1) (E(N − 1) resp.) Here we will consider that |ΨN +1 will be obtained by adding a particle in the state |α to a correlated state |ΨN (α) of N fermions Similarly the |ΨN −1 will be obtained by adding a particle in the state |a to a correlated state |ΨN −2 (a) of N − fermions These correlated states (|ΨN (α) , |ΨN −2 (a) ) are defined as two specific HTDA solutions : i) |ΨN (α) is a HTDA solution for N fermions spread over the spectrum (S) deprived of the sp state |α ; ii) |ΨN −2 (a) is a HTDA solution for N − fermions spread over the spectrum (S) deprived of the sp state |a Note that while in the above case i), one has used in the HTDA approach the N -fermion vacuum |ΦN , one has to define, in the other case ii), out of the same spectrum (S), a new vacuum corresponding to N − fermions and noted |ΦN −2 It is interesting to point out that |ΨN (α) and |ΨN −2 (a) are very similar indeed to the BCS-like state |BCSN (k) introduced in Eq (20) They are different, however, in that they correspond to an exact number of particles N (or N −2) whereas, as we have seen, |BCSN (k) does not To these two vacua |ΦN and |ΦN −2 are associated according to Eq (10) two HTDA Hamiltonians H (ν) with ν = N and ν = N − We pursue within HTDA a parallel to what is done within BCS (see Eq (20)), and make an assumption in line with the BCS quasi-particle ansatz of Eq (18) stating that |ΨN +1 = a†α |ΨN (α) , (22) |ΨN −1 = a†a |ΨN −2 (a) , (24) |ΨN +2 = a†α a†α |ΨN (α) , |ΨN = a†a aa†¯ |ΨN −2 (a) (23) (25) In the following we will denote by E (ν) (Ψ) the expectation value of H (ν) in a state January 28, 2008 10:41 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 The Higher Tamm–Dancoff Approximation 235 |Ψ When evaluating the quantities E (ν) (ΨN ±1 ) and E (ν) (ΨN ±2 ) for the approximate states defined in Eqs (22), we will further assume that Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only N −2 † N , aa = , vres , a†α = vres (26) and similar relations for the sp states |α and |¯ a However, we consider, of course, that vres acts in the space spanned, e.g., by the two Kramers degenerate states |a and |¯ a This means that in such calculations one boldly assumes that the sp states |α and |α on one hand, |a and |¯ a on the other hand are decoupled from all the other fermions Consequently, one gets the following approximate expressions for the (N + 2)-, (N + 1)-, N -, (N − 1)- and (N − 2)-fermion systems ≈ E N (ΨN +2 ) E(N + 2) E(N + 1) ≈ E N ΨN (α) + 2eα + E(N − 2) αα|vres |αα , N ≈E E(N ) E(N − 1) ≈E N −2 ≈ E (ΨN +1 ) N ΨN (α) + eα , ≈E ≈E N ΨN ΨN −2 (a) + 2ea + ≈E ≈E N −2 ≈E a¯ a|vres |a¯ a , (ΨN −1 ) ΨN −2 (a) + ea , N −2 (+) (+) (−) δ3 = = E N ΨN (α) − E N ΨN − E N −2 ΨN −2 (a) − E N −2 ΨN −2 (29) (30) ΨN −2 This leads therefore to the following expression for δ3 δ3 (28) ΨN N −2 N −2 (27) (31) (−) and δ3 : αα|vres |αα , (32) − (33) a¯ a|vres |a¯ a Finally one obtains the symmetrized shift δ as half the sum of the above evaluated (+) (−) δ3 and δ3 Quadrupole-Quadrupole Residual Interaction Let us now treat the low-multipole part of the residual interaction As an example we consider its λ = component limiting further our study to the so-called isoscalar interaction where neutrons and protons vibrate in phase This amounts to define w as w = χ2 µ=−2 Q2µ · Q2−µ , (34) acting indifferently on either charge state, i.e., yielding terms of the form Qn · Qn , Qp · Qp , Qn · Qp and Qp · Qn Phenomenologically we present here a practical way by which the strength parameter χ2 can be determined This will be illustrated by numerical calculations January 28, 2008 236 10:41 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 P Quentin et al mk (Q2 ) = n (En − E0 )k n|Q2 |0 , (35) where the sum runs in principle over all excited states |n of the Hamiltonian H and |0 denotes the corresponding ground state In Fig 1, we have reported the evolution, for the considered numerical example, 25000 40 Ca 20000 µ1(MeV.fm4) Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only performed with the Skyrme interaction (vSk ) in its SIII parameterization15 for the 40 Ca nucleus, using an estimate for χ2 taken from Ref 10 More precisely the authors of this reference showed that χ2 varies as A−7/3 To assess the reliability of our approach, we simply choose the proportionality constant to be As it will turn out, without aiming here at an optimal reproduction of data, the value retained here for χ2 , namely χ2 = −A−7/3 = −1.827 × 10−4 MeV.fm−4 , may be deemed to yield not completely unrealistic results The property which can be used for a fit is related to the distribution of quadrupole strength in the nuclear spectrum reflected in its various moments SIII L=2 15000 10000 5000 0 20 40 60 E(MeV) 80 100 Fig Variation of the contribution µ1 (in MeV.fm4 ) to the moment m1 (Q2 ) as a function of the excitation energy E (in MeV) of the contribution µ1 (n) of the state |n to the moment m1 (Q2 ) To guide the eye µ1 (n) has been artificially broadened, so the plot actually represents µ1 (E) = n µ1 (n) √ (E − En )2 exp − 2σ 2π σ , (36) with σ = 0.6 MeV Incidentally we note that the isoscalar quadrupole resonance is found within the right energy range (of the order of 20 MeV) The numerical challenges which are met in such a calculation are related to the necessary truncation inherent to our HTDA calculations First, in this calculation we have limited the space of many-body excitations merely to 1p − 1h states involving January 28, 2008 10:41 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 The Higher Tamm–Dancoff Approximation 237 sp states lying in an interval defined by a cut-off parameter X such that e p , eh λ+X , (37) where λ is the Fermi level defined as half the sum of the last occupied and first unoccupied sp states As a matter of fact a smooth boundary cut-off described in Ref has been considered, but this is somewhat irrelevant for the present discussion For a given value of X, one may check the adequacy of the retained sp space upon assessing to which extent the sum-rule theorems, equating moments with quantities involving only expectation values in the correlated ground state, are satisfied In practice we have considered the energy weighted sum rule (EWSR) m1 (Q2 ) = A m 0|r2 |0 + 0|Q20 |0 , (38) where m is the nucleonic mass and A the total number of nucleons This is illustrated in Fig where the percentage of the EWSR as a function of the sp cut-off X is presented For each value of X we have retained the same number of excitated states |n entering Eq (36), namely 100 here, which corresponds to a maximal excitation energy of about 95 MeV (see discussion below of Fig 4) For example 75 70 40 65 SIII Ca L=2 60 m1(%) Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only λ−X 55 50 45 40 35 30 14 16 18 20 22 24 X(MeV) 26 28 30 32 Fig Moment m1 as a function of the cut-off parameter X of the sp spectrum expressed as a percentage of the “exact” value given by the right-hand side of Eq.(38) the “exact” value of the m1 moment for X = 30 MeV as determined by Eq (38) is about 75000 MeV.fm4 A reasonable account of the sum rule is obtained above X ≈ 24 MeV for which one reaches a plateau Actually, at the scale of Fig 2, this plateau is slightly sloping down as X increases This very slow decrease of m (about 0.6%/MeV) could be due to the corresponding opening of more 1p − 1h excitations Indeed the latter may contribute more and more to the excited states |n lying above the highest one taken into account At this stage, we can consider January 28, 2008 10:41 238 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 P Quentin et al that X = 30 MeV (above the start of the plateau) is a reasonable value of the sp cut-off parameter A similar pattern (increase followed by a slightly sloping down plateau as a function of X) is also observed for m−1 and m3 moments (see Fig 3) 100 3.4 3.2 80 Ca Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only m3.107(MeV3.fm4) 40 -1 m-1(MeV fm ) 90 SIII L=2 70 60 40 2.8 Ca SIII L=2 2.6 2.4 2.2 1.8 1.6 50 1.4 40 1.2 14 Fig 16 18 20 22 24 X(MeV) 26 28 30 32 14 16 18 20 22 24 X(MeV) 26 28 30 32 Same as Fig.2 for the absolute value of m−1 (left panel) and m3 (right panel) As a last test we have checked that the number of solutions |n of the HTDA secular equation for a given value of X (here X = 30 MeV) is enough to describe accurately some energies representative of the giant collective excitations, namely E1 , E3 and E5 defined by Ek = mk mk−2 (39) This is illustrated in Fig In view of the above discussion about m1 , it seems that mk (with k = 1, 3, 5) is a little bit more sensitive to the truncation of the HTDAsolutions basis than Ek This is maybe so because the latter is expressed in terms of a ratio of mk moments Therefore one has at one’s disposal the tools to make an overall fit for a series of selected nuclei on various moments of the operators acting in observed collective excitations Conclusion The HTDA calculations are currently in an exploratory phase where their practical feasibility and phenomenological relevance is assessed To be sure, the theoretical foundations of, e.g., the effective residual interactions in use would need to be grounded on more microscopic approaches Yet, at this stage, our approach seems to be able to provide a feasible and realistic alternative to currently available selfconsistent approaches (for instance HFB) while preserving an essential property, namely the particle-number conservation January 28, 2008 10:41 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2007 The Higher Tamm–Dancoff Approximation 239 E5 E3 E1 25 24 Ek(MeV) 23 40 22 Ca SIII L=2 21 20 Int J Mod Phys E 2008.17:228-239 Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY - SERIALS UNIT on 01/07/15 For personal use only 19 18 17 10 20 30 40 50 60 70 En(MeV) 80 90 100 Fig Variation of some energies representative of the giant collective excitation E (dash-dotted line), E3 (full line) and E5 (dotted line) for a cut-off parameter X = 30 MeV as functions of the maximal energy of the solutions to the HTDA secular equation Acknowledgments One of the authors (Ph Q.) acknowledges the Theoretical Division at LANL for the excellent working conditions extended to him during numerous visits Part of this work has been supported by the U.S Department of Energy under contract W-7405-ENG-36 Two others (H N and H T L.) thank the Agence Universitaire de la Francophonie for various formation programs which have made possible the present collaboration Finally the support of the IN2P3 fund for theoretical physics projects is gratefully acknowledged References 10 11 12 13 14 15 N Pillet, Ph D dissertation, U Bordeaux (2000), unpublished Ha Thuy Long, Ph D dissertation, U Bordeaux (2004), unpublished K Sieja., Ph D dissertation, U Bordeaux (2007), unpublished N Pillet, P Quentin and J Libert, Nucl Phys A697, 141 (2002) P Quentin, H Laftchiev, D Samsœn, I N Mikhailov, and J Libert, Nucl Phys A734, 477 (2004) L Bonneau, P Quentin, and K Sieja, Phys Rev C76, 014304 (2007) H Lafchiev, J Libert, Ha Thuy Long and P Quentin, in preparation B Mottelson, “The Many-Body Problem”, University of Grenoble, 1958 (Lectures at Les Houches Summer School) (Dunod, Paris, 1959), p 259; A M Lane, Nuclear Theory (W A Benjamin, Inc., New York, 1964), p.8 M Baranger and K Kumar, Nucl Phys 62, 113 (1965) L.S Kisslinger and R.A Sorensen, Mat Fys Medd Dan Vid Selsk 32, No (1960) L.S Kisslinger and R.A Sorensen, Rev Mod Phys 35, 853 (1963) D.G Madland and R.J Nix, Nucl Phys A476, (1988) W.S Myers and W.J Swiatecki, Ann Phys 81, 186 (1974) G Audi, A.H Wapstra and C Thibault, Nucl Phys A729, 337 (1988) M Beiner, H Flocard, N Van Giai, and P Quentin, Nucl Phys A238, 29 (1975)  ... in the space spanned, e.g., by the two Kramers degenerate states |a and |¯ a This means that in such calculations one boldly assumes that the sp states |α and |α on one hand, |a and |¯ a on the. .. of the theoretical foundations of the HTDA approach and in the following sections we discuss phenomenological aspects of the residual interactions used to describe pairing correlations in the. .. (−ei resp.) if the state |i is a particle (hole resp.) state with respect to the vacuum |Φ Similarly the creation operator ηi† stands for the operator a†i (ai resp.) depending on the particle