Semiclassical description of finite fermion systems at finite temperature in a generalised Routhian approach

Nuclear Physics A 764 (2006) 79–108

Semiclassical description of finite fermion systems at finite temperature in a generalised Routhian approach K. Bencheikh a , J. Bartel b,∗ , P. Quentin c,d a Laboratoire de Physique Quantique et Systèmes Dynamiques, Département de Physique, Université de Sétif,

Sétif, Algeria b Institut de Recherches Subatomiques, IN2P3/CNRS and Université Louis Pasteur, Strasbourg, France c Centre d’Etudes Nucléaires de Bordeaux Gradignan, IN2P3/CNRS and Université Bordeaux I, Bordeaux, France d T-Division, Los Alamos National Laboratory, Los Alamos, USA

Received 11 April 2005; received in revised form 3 August 2005; accepted 30 August 2005 Available online 9 September 2005

Abstract A semiclassical description at finite temperature is presented for an N fermion system experiencing velocity fields coupling to the linear momentum of each of the particles. Spin degrees of freedom are also considered. Analytical expressions are derived for various local and integrated physical quantities in the framework of the Extended Thomas–Fermi method. Low and high-temperature limits are carefully examined. This formalism is then applied to the particular case of hot rotating nuclei.  2005 Elsevier B.V. All rights reserved. PACS: 03.65.Sq; 21.10.Re; 21.60.Jz; 31.15.Bs; 31.15.Gy Keywords: Finite fermion systems; Semiclassical methods; Grand canonical approach; Collective rotations; Moments of inertia

1. Introduction The concepts and methods in use here are possibly applicable, up to minor tunings of the formalism, to diverse saturating finite fermionic systems. In particular, due to the well-known assimilation of the Coriolis pseudo-force as a Lorentz magnetic force, they would easily be

* Corresponding author.

E-mail address: [email protected] (J. Bartel). 0375-9474/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2005.08.018

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transposable to magnetic systems. For the sake of definiteness, however, our approach will be mostly developed and justified in the sole context of nuclear physics. In heavy-ion fusion reactions compound nuclei can be formed at high excitation energies and in high angular momentum (high spin) states. The compound nuclei then decays by emitting particles and gamma rays and from the analysis of these decay patterns the properties of hot rotating nuclei can be investigated. Among these properties, one may note nuclear shape fluctuations, particularly through the study of the widths of giant dipole resonance (see, e.g., [1] for a recent experimental related study). This problem has attracted in recent years the attention of some theoretical groups [2–4]. One may also mention the theoretical [5–7] and also experimental studies [8–11] concerned with the fission process, as well as the subsequent de-excitation of hot rotating fission fragments. In all cases the thermodynamical equilibrium properties of hot rotating nuclei (as, e.g., moments of inertia at finite temperature) are (or should be considered as) of paramount importance. Such properties have been studied in the general framework of finite temperature mean-field approaches, in the Strutinsky shell correction (SSC) framework [12,13], at the Hartree–Fock (HF) [14–17] or Hartree–Fock–Bogoliubov (HFB) level [18], or within a specific application of the general Landau theory of phase transition (LPT) [19]. The application of the latter thermodynamical approaches to the hot rotating case has been performed within the frameworks of the SSC in [20,21], the HFB in [22,23] and LPT theory in [24,25]. At high enough temperatures both the pairing correlations (see the discussion of the critical temperature in [26]) and the shell effects [27] should disappear. Therefore the nucleus is expected to behave like a classical liquid drop and if such a hot nucleus is brought to rotate, it could be studied in a similar manner as a rotating classical fluid which has been extensively studied at zero temperature in [28]. However, the liquid drop properties of a rotating drop have no reasons, a priori, and do not, in practice, to be the same as the ones used at zero spin [29,30]. Moreover, when applying such an approach at finite temperatures to the Helmholtz free energy, one should take into account two important modifications of the model parameters. First of all, the saturation properties should vary with the temperature (see, e.g., [31]) and furthermore the adiabatic parameters describing the collective modes, e.g., here for the global rotations, the moments of inertia should also vary upon increasing the internal excitation energy. Therefore, to take into account these effects in a microscopic theoretical description one may avoid the useless complications of the Routhian HFB calculations and merely resort to semiclassical approximations like the Extended Thomas–Fermi method (ETF) [32]. One could argue that in doing so, one makes it impossible to microscopically describe the transitions from quantal to semiclassical and/or from paired to unpaired fermionic systems. While the first remark is valid, the second is not much relevant in practice since the HFB approximation is known to be particularly unreliable in the vicinity of such a phase transition in finite fermionic systems. Now, taking into account the fact that in our approach, as in the above mentioned calculations, one uses the grand canonical approximation whose a priori justification is more practical than physical, one does not expect to describe accurately excitation regimes where a significant amount of the fermionic probability corresponds to unbound states. Thus, the window of validity of the current approach should correspond to temperatures large enough so that shell effects could be reasonably neglected and small enough with respect to the relevant fermionic separation energies. One should not worry in this context with the presence of pairing correlations since in the BCS theory, the critical temperature has been found [23] to be equal to 0.57∆0 where ∆0 is the zero temperature pairing gap. In nuclear physics pairing gaps are of the order of 1 MeV, while various microscopic calculations (see, e.g., [11,12]) have found that most shell effects should

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have disappeared around or below a temperature of 2 MeV. As a result, for stable nuclei, the relevant range could thus roughly correspond to temperatures in the 1.5–5 MeV interval. The finite temperature ETF approach has been developed by Brack and coworkers [32–34] and applied with success for the study of hot non-rotating nuclei. On the other hand, the ETF model at zero temperature has been extended to take into account the effect of the rotation within the Routhian approach using Skyrme type interactions [29,30,35–37]. The latter approach has been generalized to an arbitrary vortical velocity field [37–39]. Now the extension of the generalized Routhian approach with Skyrme type interactions at finite temperature is called for. The paper is organized as follows. In Section 2 we briefly recall how the quantum as well as the semiclassical single-particle density matrix of an N fermion system at zero temperature can be obtained and discuss its generalization, within the grand-canonical description, to an excited system at temperature T . Starting from the semiclassical h¯ expansion for the various local densities at zero temperature in the case that the associated one-body Hamiltonian contains a coupling of a collective vector field to the momentum of the particle, we derive in Section 3 the corresponding h¯ expansion at finite T . In Section 4 these densities at non-zero temperatures are used to obtain semiclassical functional expressions in the so-called Extended Thomas–Fermi method, expressing them through the local matter density ρ, first in the absence of spin degrees of freedom, and in Section 5 when taking these into account. Compact analytical forms are derived in Section 6 for various integrated quantities, such as the total energy, the Helmholtz free energy and the entropy. As an application the case of collective rotation is treated and a simple expression is found for the dynamical moment of inertia for a hot rotating nucleus. Section 7 is concerned with the high-temperature limit of the moment of inertia in the low-density case, before concluding in Section 8. 2. Semiclassical density matrices at finite temperature Let us consider a system of independent fermions whose dynamics is governed by a one-body Hamiltonian Hˆ , to be specified later, and whose eigenvectors |i (with the associated wavefunctions in coordinate space ϕi (r )) and eigenenergies εi , are defined by Hˆ |i = εi |i.

(1)

At zero temperature it is well known that the so-called canonical density or Bloch density matrix
C(r , r ; β) = ϕi∗ (r  )ϕi (r )e−βεi , (2) i

where the sum extends over the whole spectrum, is of particular interest as we will show below. Consider, for instance, a Slater determinant of N particles whose occupied states correspond to the lowest eigenstates compatible with the Pauli principle. In a somewhat ambiguous manner, for such a finite fermionic system, one may define a Fermi energy λ as any energy comprised between the energies εi of the last occupied and the first unoccupied levels. Such a pure state (in the sense of clearly having a many-body density matrix which is a projector) may be considered as the zero-temperature solution associated with the Hamiltonian Hˆ . Its one-body density matrix ρˆ at zero temperature may be written as ρˆ = Θ(λ − Hˆ ), (3) where Θ(x) is the heavyside step function. The r-representation of ρˆ
ϕi∗ (r  )ϕi (r )Θ(λ − εi ) ρ(r , r ) = i

(4)

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may be obtained through the inverse Laplace transform of the Bloch density divided by β: 

ρ(r , r

) = L−1 β→λ



c+i∞   1 r , r ; β) 1  C(  C(r , r ; β) = dβ, eβλ β 2iπ β

(5)

c−i∞

where the constant c defining the above integration path is a positive real number. What we then call the density matrix ρ(r , r ) is the result of this inverse Laplace transform for the value λ = λ of its variable such that the space integral over the local part of this density matrix yields the desired value N for the particle number, implementing thus the physical information on the particle number that is obviously lacking in the Bloch density which is only defined in the full Fock space. Through that, it clearly follows that all the single-particle properties and even (since the Hamiltonian is here purely of a one-body character) all physical properties of this zerotemperature system may be expressed in terms of the single-particle Bloch density. In order to make the study more transparent, let us for the time being ignore the spin degrees of freedom. An extension to take them into account will be presented in Section 5. The one-body Hamiltonian we will be working with is of the general form h¯ 2   +∇  · α ),  + V + h¯ ( Hˆ = − α·∇ ∇ ·f∇ 2m 2i

(6)

where V is a one-body mean-field potential, f a position dependent effective-mass form factor and the last term describes the coupling of a local (involving only the position operator) vector field α to the momentum of the particle. In the context of nuclear physics such a Hamiltonian form is encountered within the framework of the generalized Skyrme Routhian approach [39]. In that case the one-body mean-field potential V is itself a function of the local density ρ(r ) plus some additional densities, to be investigated below, and their derivatives and thus depends itself on the eigenfunctions of that same Hamiltonian, Eq. (6). We are then obviously faced with a selfconsistency problem which can be resolved by an iterative procedure. For the sake of a simple notation, we have restricted ourselves here to the case of N particles (one kind only) in a given local potential. We shall also be interested in particular to the application of our method to the case of nuclear structure calculations where one is faced with two sorts of fermions (neutrons and protons). The expressions developed below should then be applied separately for these two species. However, in self-consistent calculations, a coupling generally exists between neutrons and protons, so that the fields f , V and α for one definite charge state depend on the properties of the solutions for the other nucleon state as well. Once the single-particle density matrix is known, other important quantities can be derived from it, such as the kinetic-energy density matrix r · ∇  r  )ρ(r , r ) τ (r , r ) = (∇

(7)

and the current density matrix 1   r  )ρ(r , r ). −∇ j(r , r ) = (∇ 2i r

(8)

A semiclassical expansion of the density matrix can easily be obtained at this level through the so-called Wigner–Kirkwood (WK) h¯ expansion of the Bloch density matrix [40,41]. Before proceeding, let us make the following general remark. Instead of expressing all relevant local

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quantities in terms of the mean-field potential V it turns out to be much more profitable, as we shall see, to work with an effective potential Veff (r ) = V (r ) −

m α2 . 2f

(9)

As a result rather simple and transparent forms are obtained as compared to the much more cumbersome expressions given by the authors of Ref. [42] as shown in Section 3. The fact that one is able to express all relevant quantities in this simple way in terms of an effective potential is due to the property that for a Hamiltonian of the form (6) one can restore the spherical symmetry . in momentum space through the trivial transformation [35,42] of the momentum p → p + m fα In terms of the effective potential we can now give the WK expansion of the Bloch density matrix as   ∞
(WK)  (TF)  n  (r , r ; β) = C (r , r ; β) 1 + (10) C h¯ χn (r , r ; β) , n=1

where CTF is a generalisation to the case α = 0 of the zero-temperature Thomas–Fermi (TF) approximation, whose phase-space representation is    p) CTF (R,  = e−βHw (R,p) ,

(11)

 p) where HW (R,  is the Wigner transform of the Hamiltonian Hˆ given in Eq. (6). Through an inverse Wigner transform [40,43,44] one obtains  s; β) = Nβ e C (TF) (r , r ; β) = C (TF) (R, with Nβ defined by  3/2 m Nβ = . 2π h¯ 2 fβ

m  s 2 −i h¯mf α ·s −βVeff (R)− 2h¯ 2 fβ

e

(12)

(13)

Here we have introduced the center-of-mass and relative coordinates R = 12 (r + r ) and s = r − r and the functions χn (r , r ; β) contain powers of β and gradients of the single-particle effective  up to nth order (see, e.g., [45]). To be very specific, one could have added potential Veff (R) the single-particle potential V (or Veff ) explicitly, when writing the Bloch density, starting in fact with Eq. (2), since the eigenfunctions ϕi (r ) and the eigenenergies εi of the one-body Hamiltonian are, obviously, obtained for that potential. One would have then written the r.h.s. of Eq. (10) in the form C(r , r , Veff ; β), thus emphasizing explicitly the dependence on the effective potential. Such a notation is, however, not the one in common use and we will mostly rather use the traditional form as already done in Eqs. (2)–(12). To obtain closed forms for quantities at finite temperatures we will, however, have recourse to such a notation in the following. Notice that it is the phase factor (linear in s ) appearing in the TF Bloch density, Eq. (12), which generates, in the presence of the vector field α , a non-vanishing current density which otherwise would be zero, as can be easily verified. Starting from the WK expansion of the Bloch density, Eq. (10), the corresponding WK expansion of density matrix ρ (WK) (r , r ) is easily obtained through Eq. (5). For finite temperature, T > 0, it was shown in Ref. [13] that the corresponding Bloch density matrix CT (r , r ; β) is simply given as the product CT (r , r ; β) = C(r , r ; β)g˜ T (β),

(14)

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where C(r , r ; β) is the “cold” (i.e., zero temperature) Bloch density and g˜ T (β) the two-sided Laplace transform of the function gT (ε) =

1 ε 4T cosh2 ( 2T )

,

(15)

or specifically (we put the Boltzmann constant equal to unity and measure the temperature in energy units)

g˜ T (β) = Lε→β gT (ε) =

+∞ e−βε gT (ε) dε =

−∞

π βT . sin(πβT )

(16)

Exactly as in Eq. (5), the density matrix at finite temperature can then be written as 

ρT (r , r

) = L−1 β→λ



c+i∞   1 C(r , r ; β) 1  e βλ CT (r , r ; β) = · g˜ T (β) dβ. β 2iπ β

(17)

c−i∞

The reason why the density matrix at finite temperature can be expressed in this simple form stems from the so-called convolution theorem for Laplace transforms. Indeed, writing the density matrix ρT (r , r ) in the form
ϕi∗ (r  )ϕi (r )ni (T ), (18) ρT (r , r ) = i

where the ni (T ) are the Fermi occupation numbers ni (T ) =

1

(19)

1 + exp( εiT−λ )

one finds [13] that Eq. (18) (and in particular with the above occupation number ni (T )) is simply obtained by convoluting the zero temperature density matrix with the temperature smoothing function gT (ε) of Eq. (15). To obtain a semiclassical expression of the Bloch density matrix at finite temperature we will use the above introduced WK expansion by inserting Eq. (10) into (14) and using (16) which then takes the form +∞ dε

(WK) (r , r ; β) = Nβ CT

−∞



× 1+

e

−i h¯mf α ·s



ε 4T cosh2 ( 2T )

∞


e−β[ε+Veff (R)] e

−

m s2 2h¯ 2 β





h¯ χn (r , r ; β) , n

(20)

n=1

where the Laplace-transform term e−βε adds up to the exponential involving the effective potential in the TF Bloch density. As mentioned above the functions χn depend only on the gradients  evaluated at the center-of-mass coordinate R = 1 (r + r ), which of the effective potential Veff (R) 2 allows us to rewrite (20) in the following closed form (WK) (r , r ; β) = CT

+∞

−∞

dε 4T cosh

2

ε ( 2T

)

  + ε; β , C (WK) r, r , Veff (R)

(21)

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where the notation C (WK) (r , r , Veff + ε; β) merely indicates a shift by ε of the effective singleparticle potential. It is now easy to show that what holds for the Bloch density does also for the density matrix which is simply obtained by inserting Eq. (21) into (17). One then may write with obvious notation ρT(WK) (r , r ) =

+∞

−∞

dε 4T cosh

2

ε ( 2T

)

  +ε . ρ (WK) r, r , Veff (R)

(22)

If one is also interested in the WK expansions at finite temperature of the kinetic energy and the current density matrices these are easily obtained from Eqs. (7) and (8) upon using Eq. (22) (WK) τT (r , r ) =

+∞

dε 2

−∞

4T cosh

ε ( 2T

)

  +ε τ (WK) r, r , Veff (R)

(23)

  +ε j(WK) r, r , Veff (R)

(24)

and (WK) (r , r ) = jT

+∞

−∞

dε 4T cosh

2

ε ( 2T

)

using the same notation as above. Throughout this work we shall be particularly interested in the local parts of the previous density matrices, namely the matter density ρT (r ), the kinetic energy density τT (r ) and the current density jT (r ), but also some other densities relevant for the finite-temperature case. Besides such local densities, global or integrated quantities are also of interest for the description. At finite temperature the relevant global quantity turns out to be the Helmholtz free energy which is defined as F = E − T S,

(25)

where E is the total internal energy and S the entropy of the system. In the independent particle picture, E and S are respectively given in terms of the single particle energies εi and their respective occupation numbers ni , as
εi ni (26) E= i

and S=−




ni n ni + (1 − ni ) n(1 − ni ) .

(27)

i

Following Ref. [34], one may introduce the local entropy density σ (r ) through  S = σ (r ) d 3 r with σ (r ) = −






ϕ (r ) 2 ni n ni + (1 − ni ) n(1 − ni ) , i i

(28)

(29)

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where the ϕi are the eigenfunctions of the one-body Hamiltonian Hˆ . Upon using the Schrödinger equation (1) one can express in this independent particle picture the total internal energy as    2 h¯ τ (r ) + V (r )ρT (r ) d 3 r E= (30) 2m T and what we will call the total internal (generalised) Routhian [39] as    2 h¯  α ( r ) · j ( r ) d 3 r. R= τT (r ) + V (r )ρT (r ) + h ¯ T 2m

(31)

Inserting Eqs. (28) and (30) into the definition equation (25) one can define a local free energy density F(r ) =

h¯ 2 τ (r ) + V (r )ρT (r ) − T σ (r ) 2m T

(32)

 so that F = F(r ) d 3 r. To obtain the semiclassical h¯ expansion for the free energy density it is more convenient to relate it to the local Bloch density matrix and is explicitly given [13] by   1 −1 F = λρT (r ) − Lβ→λ 2 CT (r ; β) , (33) β where λ here is the chemical potential. As a consequence, the h¯ expansion of the entropy density σ (r ) can be deduced from Eq. (32), since the semiclassical expansions for the densities τ (r ) and F(r ) are known through Eqs. (23) and (33), respectively. 3. Semiclassical h ¯ expansion for local densities In this section explicit h¯ expansions at zero and finite temperature will be presented for the various local densities ρ(r ), j(r ), τ (r ) and, in the case of finite temperature, for F(r ) and σ (r ). In order to obtain the densities at finite temperature, it is necessary to know their expressions at T = 0 as shown in the previous section. Given the Hamiltonian form, Eq. (6), Grammaticos and Voros [42] derived in the zero-temperature case semiclassical expansions up to order h¯ 2 for the above mentioned densities (note that, as already mentioned, spin degrees of freedom will not be considered here, but their implementation will be dealt with in Section 5). 3.1. The h¯ expansion at zero temperature The first semiclassical study of fermionic rotational properties has been performed  years ago by Brack and Jennings [46]. They have calculated the partition function Z(β) = C(r , β) d 3 r for a rotating N particle system in some given local model potential in the absence of a spin–orbit coupling term and with an approximate (TF) treatment of the variable effective mass. Here, we deal, of course, with a more general problem and aim to extend it to finite temperatures. In this context we first briefly recall the zero-temperature results for the expressions of various local densities in terms of the effective potential Veff of Eq. (9), in a form suitable for their later extensions. In particular it is interesting to note that at this level we do not use the Wigner–Kirkwood h¯ expansion of the Bloch density, Eq. (10), to obtain these local densities, but simply take directly the semiclassical expansions obtained by Grammaticos and Voros [42] using the method of Baraff and Borowitz [47] which leads directly to an h¯ expansion of the density

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87 ˆ

matrix ρˆ = Θ(λ − Hˆ ) without going through the use of the Bloch density operator e−β H . Subsequently we will, however, rather establish the Wigner–Kirkwood expansion of the Bloch density for the case of the Hamiltonian (6), not given in [42], since such an expansion will be needed for the calculation of the Helmholtz free-energy density, Eq. (33), using Eq. (14). We begin with the local particle density which in terms of Veff can be written as   1 3 1 m2 m2  −1 2 −3 Ak , (34) ρ(r ; λ) = k + + Bk − ( ∇V ) k F eff F 3π 2 F 24π 2 h¯ 2 f 2 F h¯ 4 f 2 where



h¯ kF (r ) =

2m λ − Veff (r ) f

(35)

is the local Fermi momentum in terms of the effective potential Veff and where, similarly to Ref. [34], we have defined   2  2f 7 ∇f ∇ , A= −2 4 f f 2 2 f 2 1   2 α + ∇  · (  α  eff − ∇ α + α · ∇ B = −2 ∇ Veff + ∇f α · ∇) · ∇V m m 2  · ∇)  α  · (  α (  )2 α (∇f ∇f α · ∇) α × ∇f +2 . (36) −2 − f f f2 For the kinetic energy density we find in the same way   1 5 m2 ρ 2 1  2 1 m2  3   eff )2 A 3(∇V τ (r ; λ) = k + α  + ρ + k + B k + ∇ F F 4 5π 2 F h¯ 2 f 2 24π 2 h¯ 4 f 2    4m  1  −1  (∇Veff × α ) · ∇ × α − (∇f × α ) kF , − f f

(37)

where   2  2f 25 ∇f 5∇ A = , − 12 f 3 f  2 2 f 2 m2 5   2 α − ∇  · (  α  eff + ∇ α + 3 B = 2 −4 ∇ α·∇ Veff + ∇f α · ∇) · ∇V 2 m m 2 h¯ f   · ∇)  α − 4  ∇  · α ) + 1 −8  2f  2 f + 4( α · (∇f α2∇ − 4( α · ∇)( α · ∇) f  )2  ( α × ∇f     , + 4( α · ∇f )(∇ · α ) + 4∇f · ( α · ∇) α +7 f2 

(38)

and for the current density we get     × α  −1  mρ m  1 m  ∇f   j (r ; λ) = − kF , (39) α + A kF − 2 ∇Veff × ∇ × α − f hf 12π 2 h¯ f h¯ f ¯ where

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 1 2  2      · ∇)  α − (∇  · α + α · ∇)  ∇f  ∇ f + (∇f A = ∇(∇ · α ) − ∇ α + f  1 3   × α ).  − ∇f × (∇ × α ) + ∇f × (∇f 2 2f 2

(40)

From the very definition of the local Fermi momentum, Eq. (35) it is evident that the above h¯ expansions for ρ, τ and j are limited to the classically allowed region (V (r ) < λ) which could be explicitly taken care of by multiplying their expressions by a heaviside step function Θ(λ − Veff (r )). The reader might find our semiclassical expressions still very involved and we will show in the following, how to further reduce them substantially. Let us mention, however, that already at this level we have reduced the number of terms appearing in the Grammaticos–Voros second order expressions quite a bit (passing from 18 terms to 11 in the expression for ρ, from 27 terms to 17 for τ and from 21 terms to 10 for the current density j). Let us now turn our attention to the local Bloch density C(r ; β). As pointed out above, its h¯ expansion was not given in [42], but we can, obviously, obtain the local part of C(r , r ; β) simply by inverting Eq. (5)

C(r ; β) = βLλ→β ρ(r ; λ) = β

∞

e−βλ ρ(r ; λ) dλ.

(41)

0

Inserting the expression of the particle density, Eq. (34), into the above equation and upon using the following property of Laplace transforms  ν  (ν + 1) −βV (r ) e (42) Lλ→β λ − V (r ) Θ λ − V (r ) = β ν+1 we then obtain the following form for the local Bloch density    3/2

m h¯ 2 −βVeff (r ) 2 2 3  C(r ; β) = 1+ Afβ + Bmβ + f (∇Veff ) β . (43) e 24m 2π h¯ 2 β We have deliberately indicated the dependence on the Fermi energy λ of the local densities ρ(r ; λ), τ (r ; λ) and j(r ; λ) in Eqs. (34), (37) and (39), in particular to show how to calculate the corresponding Bloch density through Eq. (41). In the following it will, however, be much more convenient to return to the usual notation ρ(r ), etc., except when explicitly needed. 3.2. The h¯ expansion at finite temperature In the following we generalize to finite temperatures the Wigner–Kirkwood expansion for the various densities treated above. To find the matter density at finite temperature we insert Eq. (34) into Eq. (22) to obtain    h¯ 2 f m 3  2 . (44) AJ−1/2 − B J−3/2 + ( ∇V ) J ρT (r ) = A∗T J1/2 + eff −5/2 48mT 2f T 4T 2 Two quantities appearing in the above equation are to be defined and deserve some comments. First of all, we have introduced the so-called Fermi integrals ∞ Jµ (η0 ) = 0

xµ dx 1 + exp(x − η0 )

(µ > −1)

(45)

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89

that have been extensively studied in Refs. [48,49] (which give in particular their respective series expansions in the low and in the high temperature limits). In the context of the present semiclassical approach we encounter derivatives of these functions which necessitate their analytical continuation to values of µ < −1 for which the previous integral is not defined. We follow the prescription of Refs. [34,50] where this analytical continuation was accomplished by defining the Jµ (η) for µ < −1 by the relation Jµ−1 (η) =

1 d J (η), µ dη µ

−µ = 0, 1, 2, 3, . . . .

(46)

The argument η0 of these Fermi integrals is defined as λ − Veff (r ) (47) T which corresponds to the product of 1/T by the chemical potential of the free fermion gas locally equivalent within the local density approximation (LDA) [51,52]. It is of course a function of r through the effective potential. Since the expression of ρT (r ) is deduced in fine from the inverse Laplace transform of Eq. (17), one fixes this value of λ (as in the T = 0 case) by imposing the desired particle number, namely upon space-integration of ρT (r ). However, as will be shown later, the ETF approach, yielding a functional dependence of the relevant physical quantities as functionals of the local density, will remove the explicit dependence of our results upon λ − Veff (r ). Another quantity appearing in Eq. (44) is the coefficient A∗T with the notation of Ref. [34]. It is defined as   2m 3/2 3/2 1 ∗ T . (48) AT = 2π 2 h¯ 2 f η0 =

This factor depends on the effective mass f which makes it r-dependent. Next, the local kinetic energy density at T > 0 is readily obtained by substituting Eq. (37) into (23)   m2 ρ 2mT 1 2 1 h¯ 2 f  A J1/2 + B J−1/2 τT (r ) = 2 T α 2 + ∇ ρT + A∗T + J 3/2 2 4 8 6mT h¯ f 2 h¯ f     1  eff )2 − 4m (∇V  eff × α ) · ∇  × α ) J−3/2 .  × α − 1 (∇f 3( ∇V − f f 12mT 2 (49) In the same way, inserting Eq. (39) into (24) we obtain for the finite temperature current density jT (r )       mρ h¯ A∗T 1   × α − ∇f × α J−3/2 . (50) jT (r ) = − T α + A J−1/2 + ∇Veff × ∇ 24T 2T f h¯ f 4. ETF functionals at finite temperatures In the present section, we shall obtain the Extended Thomas–Fermi functionals for non-zero  )2 and ∇  2 V from the expressions of temperatures. As in the T = 0 case, one can eliminate (∇V  the densities τ (r ) and j (r ), Eqs. (49)–(50), on one side and from the particle density Eq. (44) on the other. Such an elimination procedure has been applied similarly in Ref. [34] for the case

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where the vector field α in Eq. (6) is absent. The remaining local densities, namely the freeenergy density F(r ) and the entropy density σ (r ) will be determined somewhat differently, as we shall see later on. To simplify the notation we shall in the following denote the ETF functionals at finite temperature by τT [ρ], jT [ρ] and F[ρ] instead of τT [ρT ], jT [ρT ] and F[ρT ]. One must bear in mind, when writing down these functionals, that in the elimination procedure between the considered quantity and the density ρ, the semiclassical h¯ correction originates not only from the second order term of that particular quantity, but has in addition a contribution from its leading term taking into account higher order corrections in ρ (see the detailed discussion of this point given for zero temperature in Ref. [45] and for finite temperature in Ref. [34]). We then obtain first the following functional expression up to order h¯ 2 for the kinetic energy density (for one kind of particles): (ETF)

τT [ρ] = τT

(TF)

[ρ; 0] + τT

[ρ; α ] + δτT [ρ; α ]

(51)

upon splitting τT [ρ] into three contributions: • the ETF kinetic energy density (up to order h¯ 2 ) in the absence of the vector field α   2 5 2mT (∇ρ) (ETF)  2ρ + − 3ζ ∇ τT [ρ; 0] = 2 A∗T J3/2 (η) + (ζ + ν) ρ 12 h¯ f   2    2 7 9 ∇ f 7 9 (∇f ) + − ζ ρ + − + (ζ + ν) ρ 24 2 f 48 4 f2    7 9ζ ∇ρ · ∇f + + 3ν − ; 24 2 f

(52)

• the TF contribution due to this vector field (TF)

τT

[ρ; α ] =

m2 ρ h¯ 2 f 2

α 2 ;

• the h¯ 2 correction generated by the vector field α     mA∗T 3 9  2 α + 1 + 9 ζ ∇  2 α 2 J−1/2 (η) − ζ α · ∇ δτT [ρ; α ] = 6f T 8 2 16 4   1 9  ∇  · α )  · (  α − 1 ( + ζ ∇ α · ∇)( − α · ∇) 8 2 2 1  · ∇)  α + (  2 f + (  )(∇  · α ) + −2 α · (∇f α · ∇) α · ∇f 2f    )2

α × ∇f  · (  α + 7 − 9 ζ (  2 f + ∇f − α 2 ∇ α · ∇) 8 2 f2     ∇ρ  × α ) .  × α − 1 (∇f + 36 ζ × α · ∇ 6ρ f

(53)

(54)

In the above, ζ and ν are position dependent functions (through their dependence on the variable η) ζ (η) = −

1 J1/2 (η)J−3/2 (η) 2 12 J−1/2 (η)

(55)

K. Bencheikh et al. / Nuclear Physics A 764 (2006) 79–108

91

and 2 3 J1/2 (η)J−5/2 (η) 3 ν(η) = − ζ (η) + 36ζ 2 (η) − . 3 2 8 J−1/2 (η)

(56)

Notice that these two functions which determine the temperature dependence of the above functionals can be expressed entirely in terms of the Fermi integrals J1/2 (η) and J−1/2 (η) and their first and second derivatives, since   1 d 1 (57) ζ (η) = − J1/2 (η) 6 dη J−1/2 (η) and ν(η) = −3

J1/2 (η) dζ 3 dζ /dη =− . J−1/2 (η) dη 2 d( n J1/2 )/dη

(58)

In the preceding equations the new variable η is not to be confused with the variable η0 given explicitly in Eq. (47). It is, in fact, defined (see Ref. [34]) through ρT (r ) = A∗T J1/2 (η).

(59)

To better understand the difference between η and η0 let us recall that the full semiclassical expansion of the density ρ(r ) is given by Eq. (44) which, when comparing with (59) allows us to write the following relationship between η and η0  h¯ 2 f m AJ−1/2 (η0 ) − B J−3/2 (η0 ) J1/2 (η) = J1/2 (η0 ) + 48mT 2f T  3  2 + ( ∇V ) J (η ) (60) eff −5/2 0 . 4T 2 Neglecting the h¯ 2 correction in the above equation, one is trivially left with η = η0 since J1/2 (η) is a strictly monotonic function. Including the later term one can formally write η = η0 + η 2 + · · · ,

(61)

where η2 is of order h¯ 2 and can be explicitly extracted from the following equation obtained from a Taylor expansion of the function J1/2 (η) around η = η0 , i.e.,  1 (62) J1/2 (η) = J1/2 (η0 ) + (η − η0 )J−1/2 (η0 ) + O h¯ 4 , 2 where we have used the recursion property of the generalized Fermi integral given in Eq. (46). Confronting Eqs. (62) with (60) one obtains together with (61) the following expression for η2   h¯ 2 f m 3  2 AJ−1/2 (η0 ) − B J−3/2 (η0 ) + η2 = (∇Veff ) J−5/2 (η0 ) . (63) 24mT J−1/2 (η0 ) 2f T 4T 2 Comparing the result obtained for the ETF functional τT(ETF) [ρ; 0], Eq. (52), in the absence of the vector field α with the one given in Refs. [32,34] the interested reader could object that the h¯ 2 corrections have locally a different analytical form. One may show [53], however, that they have the same limit as T → 0 and they integrate, moreover, analytically to the same total kinetic energy functional. The reason for this difference has its origin in the different way in which the

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functional τT [ρ] has been determined in Refs. [32,34] as explained more in detail at the end of this section. In a way similar to the introduction of an effective potential Veff through Eq. (9) to simplify the calculation, we have found that using fα rather than α casts the expression of δτT [ρ; α ] in Eq. (54) in a much simpler form, since it can, indeed, be written as     mA∗T f 2 α × ∇  × α δτT [ρ; α ] = − · ∇ 6f T 2 f f       2  α 1 9   × α + f 2 1 + 9 ζ ∇  × α + ζ × ∇f · ∇ − 2f 8 2 f f 8 2 f     f2  α  × α · ∇ J−1/2 (η) + 36ζ (64) ∇ρ × 6ρ f f which can still be simplified to take the very compact form     mf 1/2 A∗T  · f 1/2 J−1/2 (η) α × ∇  × α δτT [ρ; α ] = ∇ 12T f f 2   3  × α . − (1 − 12ζ )f 1/2 J−1/2 (η) ∇ 4 f

(65)

Let us mention at this point that for a Hamiltonian of the form (6), the kinetic energy density only occurs in combination with the effective mass, i.e., as f (r )τ (r ). Using the above expression, one finds that f δτT [ρ; α ] has, in particular, the nice property that upon integration the first term in (65) vanishes, since f 3/2 A∗T is a position independent factor (see Eq. (48)), which was observed before at zero temperature in Ref. [54]. The corresponding functional expression found for the current density at finite temperature is given by    α 2 h¯ A∗T mρ  ∇  · α ) − ∇  2 α + ∇ f α + (∇f · ∇) J−1/2 (η) ∇( α + jT [ρ] = − 24T f f hf ¯        ( α · ∇)∇f 1 ∇f × (∇ × α ) (∇ · α )∇f − + 18ζ − − f f 2 f       × α ) ∇ρ 3 ∇f × (∇f  × α ) .  × α − 1 (∇f − 18ζ + 36ζ × ∇ (66) + 2 f 3ρ f Here again the above expression can be substantially simplified by introducing, as above, the natural field α /f . We thus obtain      h¯ A∗T mρ (36ζ + 1)  × ∇  × α  × α + f ∇ J−1/2 (η) jT [ρ] = − α + ∇f × ∇ 24T 2 f f hf ¯    ∇ρ  × α × ∇ (67) + 36ζf 3ρ f yielding the still more compact form

   h¯ f 1/2 A∗T mρ α 1/2    jT [ρ] = − . α + ∇ × f J−1/2 (η) ∇ × 24T f hf ¯

(68)

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93

Let us now come to the expression of the free-energy density functional. We calculate it through Eq. (33) using Eq. (14) together with the h¯ expansion of the zero-temperature Bloch density given in Eq. (43). As a result, applying the same elimination procedure as above, the WK expansion of the free energy density is    2  2 9 (∇ρ) (∇f ) 2 h¯ 2 7 F[ρ] = T ηρ + Veff ρ − T A∗T J3/2 (η) + ζf ρ + ζ− 3 2m ρ 4 48 f   f 2 1 2 5   − ∇ ρ + ρ ∇ f + 3ζ − ∇ρ · ∇f 6 6 12  2 2 mA∗T J−1/2 (η) ∇ α  2 α + ∇  · (  α − + α · ∇ α · ∇) − 24T 2  · ∇)  α  · (  α (  )2  α (∇f ∇f α · ∇) α × ∇f +2 (69) −2 − f f f2 which can be written as

   2  2 9 2 h¯ 2 (∇ρ) 7 (∇f ) ∗ F[ρ] = T ηρ + Veff ρ − T AT J3/2 (η) + ζf + ζ− ρ 3 2m ρ 4 48 f        ∗ 2  2f mf AT J−1/2 (η) f 2 ∇ 5  α 2   − . ∇ ρ −ρ + 3ζ − ∇ρ · ∇f − ∇× 6 f 12 24T f (70)

When using the approach of Grammaticos and Voros, Ref. [42], the coefficient in front of the  2 f in Eq. (69) is found as −1/12 rather than +1/6. One has to remember, however, that term ρ ∇ their mean-field potential VGV is related to the usual potential V by the relation h¯ 2  2 (71) ∇ f 8m and one recovers the usual coefficient +1/6 mentioned above as given in [32]. When calculating 2  2 ρ − ρ ∇ f ) clearly vanishes after partial integration. the total free energy, the term − f6 (∇ f At the end of Section 2 the h¯ expansion of the entropy density σ (r ) has been obtained through the use of Eq. (32). One may proceed in a similar way to get the functional σ [ρ]. Let us notice, however, that the entropy density deduced in this way is not unique and has, in fact, been determined differently in Refs. [32,34], namely through the following local version

∂F

(72) σ =− ∂T ρ VGV = V +

of the thermodynamical relation S = −∂F /∂T |V , where the free energy density F(r ) was determined through Eq. (33). This is the reason why we find an entropy (and kinetic energy) density, which have no reason to be the same as those given in Refs. [32,34]. They, however, need to integrate to the same respective global results, which is, indeed, the case. Let us show that the above functionals have the correct zero-temperature limits as they should. Upon using the relevant limits for ζ (η) and ν(η) (see Refs. [32,34]) as well as the limit  ∗  AT J−1/2 (η) 6m(3π 2 )−2/3 ρ 1/3 lim = (73) T →0 T h¯ 2 f

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which is easily obtained from the asymptotic form of the Fermi integrals, all the functional expressions for τT [ρ], jT [ρ] and F[ρ], and in particular their contribution for α = 0, reduce to their expressions known for the zero-temperature case [35]. 5. ETF spin dependent functionals A complete treatment of the theory should take into account the spin degrees of freedom of the particles. For this purpose, one has to consider two additional local densities, namely the spin vector density ρ(  r ) (sometimes written as [55] s(r )) and the spin current tensor density Jµν (r ) [55]. All of these densities may be formally introduced using the density operator ρˆ defined below by its matrix elements in coordinate and spin space
ˆ r , σ  = nk ϕk (r , σ )ϕk∗ (r  , σ  ), (74) ρ(r , σ ; r , σ  ) = r , σ |ρ| k

where ϕk (r , σ ) = r , σ |k is the single-particle wavefunction associated to the eigenstate |k (more precisely the component of the corresponding spinor) and nk is the occupation number corresponding to that state. In addition to ρ(r , σ ; r , σ  ) we can define the current density matrix as 1    )ρ(r , σ ; r , σ  ). j(r , σ ; r , σ  ) = (∇ −∇ (75) r 2i r We may expand now the density matrices (74) and (75) in terms of the Pauli spin matrices and since one is only interested here in the local parts in coordinate space of these quantities, one will merely consider 1 1 σ |σ   ρ(r , σ ; r, σ  ) = ρ(r )δσ,σ  + ρ (r ) · σ | 2 2

(76)

1 1

j(r , σ ; r, σ  ) = j(r ) δσ,σ  + Jµν (r )eµ σ |σν |σ  , 2 2

(77)

and 3

3

µ=1 ν=1

where σν is the νth Pauli spin matrix and eµ a unit vector of the Cartesian coordinate system. We, of course, retrieve the usual matter and current density as the spin-diagonal part of the above general representation of the ρˆ and jˆ operators. Note, en passant, that using the spin current tensor Jµν (r ) one can construct from its antisymmetric part the following vector J(r ) =

3


eλ ελµν Jµν (r ),

(78)

λµν=1

a quantity called “spin density” in Refs. [56,57] and which we will dub here as the spin current density, to avoid confusion with the above defined spin vector density ρ(  r ). As in Refs. [35,58] we consider here the Hamiltonian h¯ 2   ×∇  ) · σ + h¯ S · σ  +∇  · α ) + 1 (W  −∇  ×W  + V + h¯ ( α·∇ (79) Hˆ = − ∇ ·f∇ 2m 2i 2i  , and a spin vector which contains the usual spin–orbit term, including a spin–orbit form factor W    field S. Both fields W and S depend only on the position r. The above Hamiltonian is a simplified

K. Bencheikh et al. / Nuclear Physics A 764 (2006) 79–108

95

version of the most general one [55] obtained from the Skyrme-type effective interaction in the Hartree–Fock approximation, when time reversal symmetry is broken. To obtain the above form for Hˆ one in fact uses an often made approximation [35,58] consisting in neglecting some small contributions in the Hamiltonian density. A part of them results from a tensor coupling between spin and gradient vectors, another implies terms involving the spin-vector kinetic energy density [55,59]. Let us now return to the semiclassical aspect of the problem by listing the h¯ expansion of the various densities or rather their increment with respect to the spin-independent case for the T = 0 case as given in Ref. [60]. For the matter density we extract from the latter reference   m2  2 kF ,  2 k −1 + 2 W (80) D δρ (r ) = F 2π 2 h¯ 2 f 2 h¯ 2 where  = S − m W  × α D (81) h¯ 2 f from which we conclude on the following form of the increment of the local Bloch density 3/2    m 2m  2 −βVeff 2 2 2 δC (r ; β) = e β + β . (82) W D h ¯ 2π h¯ 2 fβ h¯ 2 f In [60] it has also been established that the spin contribution to the kinetic energy density is   2  m2  2 − 4m α · (D  ×W  ) kF + 10 W k 3 δτ (r ) = 3 D (83) 3 h¯ 2 F 2π 2 h¯ 2 f 2 h¯ 2 f and similarly for the current density   m2 W   δ j (r ) = 2 2 D × 2 kF , (84) π hf h¯ ¯ whereas the spin-vector density and the spin-current density tensor are m  F, ρ(  r) = − (85) (h¯ D)k 2 π h¯ 2 f m m (86) W k3 . α ρ − Jµν (r ) = − h¯ f µ ν 3π 2 h¯ 2 f µν F Here αµ and ρν are the µ and ν components of the vector field α and the spin density ρ respectively and the antisymmetric second-rank tensor Wµν is defined in terms of the components Wλ  as of the spin–orbit potential W
µνλ Wλ . (87) Wµν = λ

At this level we can simplify the above expressions by taking (again) advantage of the fact that in semiclassical correction terms (beyond the TF) like the above, Eqs. (83)–(86), it is sufficient to replace all quantities by their TF approximation. That is why, due to the second-order character in h¯ of δτ as well as of Jµν , we can express the term in kF3 by its TF approximation (see Eq. (34)), i.e., by the local density ρ ρ(r ) =

1 3 k (r ) 3π 2 F

(88)

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which will, when generalized to the T > 0 case, simply become the matter density at finite temperature ρT (r ). In addition to the terms in kF3 in the above expressions we need to treat the terms linear in kF and determine their generalisation to T > 0. This is accomplished by first calculating the spin-vector density at finite temperature ρT . We proceed as outlined in Section 2, Eqs. (22)–(24), to write for the WK expansion of the (local) spin vector density at finite temperature as +∞  dε (89) ρT (r ) = ρ r, Veff (r ) + ε . 2 ε 4T cosh ( 2T ) −∞

Using (85) and performing the integration one finds A∗T J−1/2 (η0 )

 (h¯ D), (90) 2T where η0 is given by Eq. (47). But then all of the above expressions are very easily generalized to finite temperature and we obtain (for simplicity we continue to write the matter density at finite temperature as ρ instead of ρT and similarly for the other densities such as j, Jµν , etc.) m ) (91) δ j(r ) = − 2 (ρ × W h¯ f and m m Jµν (r ) = − (92) αµ ρν − 2 ρWµν h¯ f h¯ f ρT (r ) = −

and upon using the definition (78) one can write for the so-called spin-current density m 2m  J(r ) = − α × ρ − 2 ρ W h¯ f h¯ f

(93)

 , a term coupling the velocity field α to which contains, in addition to the usual term linear in W the spin vector density ρ.  In the same way we obtain for the contribution to the kinetic-energy density due to the spin degrees of freedom   ∗ 4m 5m2  2 mAT J−1/2 (η0 )  2  ×W ) . 3D − 2 α · (D + (94) δτ (r ) = 4 ρ W 4f T h¯ f 2 h¯ f The preceding equations (90)–(94) have already the desired functional form, if we recall that in the spin-vector density ρ,  Eq. (90), which is of order h¯ , the Fermi integral J−1/2 (η0 ) can be replaced by its Thomas–Fermi approximation J−1/2 (η), in the same way as what has been done for the case of the matter density (and the Fermi integral J1/2 (η)) in Section 4, where η is directly related to the local density through Eq. (59). This now allows us to write down the functional expressions at finite temperature for all of the various local densities, namely ρ,  j, Jµν , τ , F and σ including the spin contributions: ρ[ρ]  =−

A∗T J−1/2 (η) 2T

 (h¯ D),

(95)

  h¯ f 1/2 A∗T m mρ  ),  × f 1/2 J−1/2 (η) ∇  × α α + − 2 (ρ × W j[ρ] = − ∇ hf 24T f h¯ f ¯ m m ρWµν , Jµν [ρ] = − α ρ − h¯ f µ ν h¯ 2 f 

(96) (97)

K. Bencheikh et al. / Nuclear Physics A 764 (2006) 79–108

τ [ρ] =

   2  2 5 (∇ρ)  2ρ + 7 − 9 ζ ρ ∇ f + − 3 ζ ∇ ρ 12 24 2 f h¯ 2 f   2     9 (∇f ) 9ζ ∇ρ · ∇f 7 7 + 3ν − + − + (ζ + ν) ρ + 48 4 24 2 f f2     2m2  2 m2 ρ 2 mf 1/2 A∗T  α  × α + 4 ρW + 2 α + ∇ · f 1/2 J−1/2 (η) × ∇ 12T f f h¯ f 2 h¯ f 2 2   3  × α − (1 − 12ζ )f 1/2 J−1/2 (η) ∇ 4 f    mA∗T J−1/2 (η) 3 2m 2    + − 18ζ D − 2 α · (D × W ) . 2f T 2 h¯ f 2mT

97

A∗T J3/2 (η) + (ζ + ν)

(98)

Leaving out terms whose contributions vanish under spatial integration, the total free energy density is then given as    2  2 9 (∇ρ) 7 (∇f ) 2 h¯ 2 F[ρ] = T ηρ + Veff ρ − T A∗T J3/2 (η) + ζf + ζ− ρ 3 2m ρ 4 48 f      2 mf 2 A∗T J−1/2 (η) 5   −  × α + 3ζ − ∇ρ · ∇f ∇ 12 24T f −

∗ 1 2m  2 AT J−1/2 (η)  2 − ρ W (h¯ D) . 2 h¯ 2 f 4T

(99)

 ) in the above expression has kept exactly the same form The spin–orbit term (quadratic in W obtained in [32,34] in the absence of the vector field α . It is interesting here to point out that this vector field enters (quadratically) via the definition of the effective potential Veff . In addition,  × ( it has the simple effect of generating two h¯ 2 corrective terms, one (quadratic in ∇ α /f )) representing the coupling of the vector field α to the orbital motion and the other (quadratic  including its coupling to the spin along with the spin field S.  Notice, however, that in a in D) selfconsistent description the vector field α contains so-called Thouless–Valatin selfconsistency terms originating from both the orbital motion and the spin degrees of freedom [55,61]. Let us close this section by the following remark. As can be seen from the expressions of the various density functionals, Eqs. (95)–(99), these are expressed in part through terms involving A∗T J3/2 (η) and A∗T J−1/2 (η). It is possible, however, to express such factors in terms of more transparent physical quantities, namely the local density ρ and the derivative of the logarithm of the Fermi integrals J1/2 (η) and J3/2 (η). Indeed, upon using the definition of the density ρ(r ), Eq. (59), and the derivative of the Fermi integrals, Eq. (46), one obtains  

−1 d 3 n J3/2 (η) A∗T J3/2 (η) = ρ 2 dη

(100)

and in the same way A∗T J−1/2 (η) = 2ρ

d n J1/2 (η) . dη

(101)

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K. Bencheikh et al. / Nuclear Physics A 764 (2006) 79–108

6. Integrated quantities in the generalized Skyrme Routhian approach With the use of the finite-temperature ETF local functionals obtained above, one can now  3 r and the total Helmholtz free calculate global quantities such as the total energy E = E( r ) d  energy F = F(r ) d 3 r and deduce there from the entropy S. The energy density E(r ) for a Skyrme type effective interaction in the absence of time reversal symmetry is a functional of the local densities ρq , τq , jq , ρq and Jq . As already mentioned above, we continue, to simplify the notation, to write the different densities at finite temperature without the subscript T . The energy density can then be written explicitly in the form [35,58] E(r ) =


h¯ 2
fq (r )τq (r ) + V(r ) − B3 j 2 (r ) − B4 jq2 (r ) 2m q q  
 q  + − B9 J(r ) · ∇ρ Jq (r ) · ∇ρ q



 × j) + + B9 ρ(  r ) · (∇




  × jq ) ρq (r ) · (∇

q

 + B10 ρ + 2




B11 ρq2

+ B12 ρ ρ γ

2

γ + B13 ρq

q




 ρq2

,

where V(r ) = B1 ρ + B2 2


q

ρq2

(102)

q

2

+ B5 ρ ∇ ρ + B6




 2

ρq ∇ ρq + ρ

q

γ

B7 ρ + B8 2




 ρq2

(103)

q

is that part of the potential energy which depends exclusively on the density ρ and its gradients. Here and throughout the remaining part of this section, any  density function which appears without charge index refers to the total density, as, e.g., ρ = q ρq . We have also adopted, for more transparency, the notation of Refs. [35,58] expressing the energy density through the constants Bi , instead of the usual Skyrme force parameters (see, e.g., Table 1 of Ref. [35]). The parameter γ in the two above equations characterises the density dependence of the effective interaction [62]. Let us now assume that we are dealing with a problem constrained on both the momentum and the spin. Namely we will perform a variational calculation, where the expectation value of the generalized N -body “Routhian” Hˆ  = Hˆ −

N


1 ˆ  · σi α 0 (ri ) · pˆ i + pˆ i · α 0 (rˆ i ) + Ω 2

(104)

i=1

ˆ Carrying out the variais to be minimized rather than the one of the many-body Hamiltonian H. tion then leads to an eigenvalue problem with the one-body Hamiltonian Hˆ given in Eq. (79). As discussed in Refs. [39,63] such a Routhian allows for the description of various collective modes (or their combination) like

K. Bencheikh et al. / Nuclear Physics A 764 (2006) 79–108

99

(a) the global rotation defined by the velocity field1  × r, α 0 (r ) = Ω

(105)

where, as well known, the constant vector 1 = ∇ (106) × α 0 Ω 2 acts also as a Lagrange multiplier for the spin degree of freedom; (b) a vortical intrinsic motion with a velocity field linear in r à la Chandrasekhar [64] with ˆ α 0 · pˆ = ω  · K,

(107)

ˆ is the so called Kelvin circulation operator (see Ref. [63] where ω  is half the vorticity of α 0 and K for details). Note that in this case no related effect on spin degrees of freedom is considered; (c) finally as stated in the introduction this formalism is suited for the description of a system of electrons subject to a uniform magnetic field B where α 0 (r ) = −

|e|  (B × r) 2m

  × α 0 = − |e| B.  =∇ and Ω m

(108)

Evaluating now the  expectation value of the Routhian, as defined in Eq. (104), we find together with the identity j2 = q j · jq the following expression for the energy density in the laboratory frame E(r ) =


h¯ 2
h¯
 q (r ) · Jq (r ) fq (r )τq (r ) + V(r ) + α q (r ) · jq (r ) + W 2m q 2 q q +

1
 h¯
h
 q · ρq (r ) + ¯ Ω α (r ) · jq (r ), Sq (r ) · ρq (r ) + 2 q 4 q 2 q 0

(109)

where the form factors, already present in the one-body Hamiltonian, Eq. (79), are for a Skyrmetype effective interaction given by [35,58] • the cranking field  × (ρ + ρq ); h¯ α q = −h¯ α 0 − 2B3 j − 2B4 jq + B9 ∇

(110)

• the spin–orbit potential  q = −B9 ∇(ρ  + ρq ); W

(111)

• the spin field   × (j + jq ) − h¯ Ω . h¯ Sq = 2B10 ρ + 2B11 ρq + 2ρ γ (B12 ρ + B13 ρq ) + B9 ∇ 2

(112)

This entails that the cranking field α , as it appears in Eq. (79), is the sum of the external vector field α 0 plus the so-called Thouless–Valatin selfconsistency corrections given in Eq. (110) above. 1 Please note the opposite sign adopted here as compared to Ref. [35].

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To obtain a semiclassical approximation of the above energy density (109) we shall use the previously developed ETF functional densities for τq , jq , ρq and Jq . It is now interesting to notice that, similarly to what was done in Ref. [35] for the pure rota × r, one can derive the following relation tional case α 0 = Ω

mB
m  × ρq ), jq = ρ α 0 + fq δ jq − 2 9 (ρ + ρq )(∇ (113) h h ¯ ¯ q q q where δ j is the h¯ 2 correction (last two terms on the r.h.s. of Eq. (96)). Using this relation together with Eqs. (51) and (68) in Eq. (109) leads after some simple manipulations and discarding terms vanishing under space integration, to the following closed form for the energy density ET (r ) =

h¯ 2
m f (r )τq(ETF) [ρq ; 0] + V(r ) + ρ α 02 2m q 2 2
 h¯ 2 A∗(q) T fq

   1
2m ρq  2 2  J−1/2 (ηq )(1 − 12ζ ) (∇ × α 0 ) − − W 32T 2 q h¯ 2 fq q q 
 h¯ mB9 T   + Ω− (ρ + ρq )(∇ × α 0 ) + (1 − 36ζ )ρq · ρq , ∗(q) 4 2h¯ 2AT J−1/2 (ηq ) q (114) where we have introduced ηq as argument of the Fermi integral to remind the reader that this quantity is obtained by inversion of the relation (59) written now for the density (at finite temperature) ρq (r ). Similarly one must remember that in the present context one has to distinguish ∗(q) the functions A∗T of Eq. (59) for the different type of particles, which leads us to write AT in the above equation. Let us now come to the Helmholtz free energy. Starting from its expression in Eq. (99) one has to remember that a similar expression has been derived for the α = 0 case in Ref. [34] where the potential energy density is of the form V (r )ρ(r ) only in the case of an external potential V . As explained there this potential energy density becomes more complicated in the case of a density dependent effective interaction (as is the case in nuclear physics) and so-called “rearrangement terms” have to be taken into account. After some algebraic efforts one finally obtains  2
 h¯ 2 A∗(q) m T fq  × α 0 )2 J−1/2 (ηq ) (∇ F(r ) = F (α =0) (r ) + ρ α 02 − 2 48T q 
 h¯ mB9   Ω− (ρ + ρq )(∇ × α 0 ) · ρq , + (115) 4 2h¯ q where the free energy density F (α=0) (r ) in the absence of the vector field α has been given in Ref. [34]. Comparing the above expression for the free energy density F with the one for the total internal energy, Eq. (114) one directly gets for the entropy density  2
 h¯ 2 A∗(q) T fq ( α =0)  × α 0 )2 (r ) + J−1/2 (ηq )(36ζ − 1) (∇ σ (r ) = σ 2 96T q −




1

q

∗(q) 2AT J−1/2 (ηq )

(36ζ − 1)ρq2 ,

(116)

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101

where σ (α=0) (r ) is the entropy-density functional [34] corresponding to a vanishing vector field α . Let us show in the following that, as expected, the above entropy density indeed vanishes in the T → 0 limit. Here we shall focus our attention on the two additional terms in Eq. (116) generated by the velocity field α 0 , since it has been shown in Refs. [32,34] that the term σ (α=0) has the correct zero-temperature limit. Using the relevant asymptotic expansion [34] of the Fermi integral Jµ (η) and the definition (55) one easily shows that for large η values    −4 π 2 −2 1 1+ η +O η ζ (η) = , η 1. (117) 36 3 Since in the limit T → 0 the parameter η goes to infinity like 1/T one concludes that the term ∗(q) (36ζ − 1) goes to zero like T 2 . On the other hand we know from Eq. (73) that AT J−1/2 (ηq )/T 1/3

behaves like ρq /fq2 which results in the two additional terms in Eq. (116) to go to zero like T . As a simple application of the above formalism let us consider the case of thermally excited rotating nuclei. As already stated in the introduction, moments of inertia are, indeed, of capital importance in the study of equilibrium properties of such nuclear systems. Looking at the ex × r we can pression of the free energy density, Eq. (115), it is obvious that for the case α 0 = Ω  3 write the total free energy F = F(r ) d r in the form 1  2. F = F (α =0) + J Ω 2 The dynamical moment of inertia J at finite temperature to be considered, is given by  2   ∗(q)
  h¯ 2 AT fq2 h¯ 2 J−1/2 (ηq ) + − 2B9 (ρ + ρq ) χq d 3 r. J =m r⊥ ρq − 6mT 2m q

(118)

(119)

The functions χq appearing in the above equation are the so-called spin susceptibilities. In anal is proportional to the field strength B,  the ogy with magnetism (where the magnetization M magnetic susceptibility being the proportionality constant), they can be introduced through  q (r ) ρq (r ) = h¯ Ωχ

(120)

as was done at zero temperature in Refs. [35–37]. Let us now briefly explain how the χn and χp are explicitly obtained in the T > 0 case. Starting from the expression of the spin vector density ρ,  Eq. (90) and upon using the definition  q and Sq , Eqs. (81) and (112) one can write of the vector fields D ∗(q)

ρq (r ) = −

AT

J−1/2 (ηq ) 2T



 2B10 ρ + 2B11 ρq + 2ρ γ B12 ρ + B13 ρq

 × (j + jq ) − + B9 ∇

  m  h¯ Ω . × h α  − W ¯ q q 2 h¯ 2 f

(121)

Due to the first-order character in h¯ of ρq one may use the TF approximations for jq and α q in the above expression to obtain the spin vector density to leading order. To derive these TF approximations for jq and α q one can follow the same procedure as in the appendix of Ref. [35] for the case of rotation at T = 0. One then obtains for an arbitrary vector field α 0 α q(TF) = −fq α 0

(122)

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and m ρq m (123) jq(TF) = − 2 h¯ α q(TF) = 2 ρq h¯ α 0 , f h¯ q h¯ which leads to ∗(q)  AT J−1/2 (ηq ) 2B10 ρ + 2B11 ρq + ρ γ (2B12 ρ + B13 ρq ) ρq (r ) = − 2T   mB9 h¯ Ω  + 2 (ρ + ρq )(∇ × h¯ α 0 ) , − (124) 2 h¯ where we have used

  × h  + ρq ) × h¯ α 0 ) .  × j(TF) + jq(TF) = m (ρ + ρq )(∇ (125) ∇ ¯ α0 ) + ∇(ρ 2 h¯  × α = 2Ω,  one notices that the spin vector density is, indeed, Since for collective rotations ∇ 0  justifying the definition (120). Using this definition together with Eq. (125) leads parallel to Ω then to the following system of linear equations for the spin susceptibilities χn and χp   ∗(n) AT J−1/2 (ηn )

1+ B10 + B11 + ρ γ (B12 + B13 ) χn T ∗(n)

+

AT

∗(p)

AT

∗(n)

J−1/2 (ηn )  AT B10 + ρ γ B12 χp = T

J−1/2 (ηp ) 

B10 + ρ γ B12 χn

  J−1/2 (ηn ) 1 mB9 − 2 (ρ + ρn ) , T 4 h¯

T ∗(p)   AT J−1/2 (ηp )

B10 + B11 + ρ γ (B12 + B13 ) χp + 1+ T ∗(p)   AT J−1/2 (ηp ) 1 mB9 − 2 (ρ + ρp ) . = (126) T 4 h¯ Upon solving this linear system one may then calculate through Eq. (119) the dynamical moment of inertia. As an interesting by-product of our study let us investigate the temperature dependence of the spin susceptibilities which we have just determined. For this purpose we have considered the selfconsistent semiclassical solutions for the nucleus 90 Zr at various temperatures, in the range T = 0–5 MeV, and at zero angular momentum, constrained to spherical symmetry. We have used the SkM∗ Skyrme effective interaction [65]. We show in Fig. 1 the isoscalar spin susceptibility χ0 = χn + χp for different temperatures. They exhibit a Fermi-function like behaviour reflecting a volume (plus diffuse surface) character of this alignment. One also observes, as expected, a decrease of the alignment of the spin-vector density on the angular velocity with increasing temperature. One could as well define an isovector spin susceptibility χ1 = χn − χp which turns out to behave similarly, being however much smaller (around 5% of χ0 ). Inserting the corresponding selfconsistent semiclassical densities in Eq. (119), moments of inertia are obtained which may be viewed as perturbative since they do not take into account the  However, at least formally, they include the so-called possible variation of the densities with Ω.

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Fig. 1. Isoscalar spin susceptibility χ0 = χn + χp as a function of the radial variable for temperatures T = 0 (solid), T = 2 (dashed) and T = 4 MeV (dotted line).

Fig. 2. Contributions to the moment of inertia from neutrons (n) and protons (p). The rigid-body moment of inertia (1st term in Eq. (119), solid line) has been reduced by a factor 10 to fit on the same figure as the orbital contribution (2nd term in (119), dashed line) and the spin contribution (last term in (119), dotted line).

Thouless–Valatin selfconsistency terms which result from the time-odd response of the mean field to the time-odd part of the density matrix. We show in Fig. 2 the behaviour, as a function of the temperature, of the three contributions appearing in Eq. (119), namely the TF term which

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is nothing but the rigid-body moment of inertia and the two h¯ 2 corrections originating from the orbital and spin degrees of freedom. All these terms with their signs and relative sizes are consistent at T = 0 with what had been found in Ref. [35]. In can in fact be easily shown, using the asymptotic form of Eq. (73) that the different contributions to the dynamical moment of inertia in Eq. (119) tend to their corresponding terms given in Ref. [35]. The rigid-body moment of inertia of a spherical nucleus is of course proportional to the square radius of that nucleus whose (almost) quadratic variation with temperature is well known [15]. The h¯ 2 corrections (of both paramagnetic and diamagnetic type) exhibit a rather small variation at low temperatures corresponding to a slow decrease of their absolute values. 7. High-temperature (low-density) limit for the moment of inertia In Ref. [35] we have established in the zero temperature case the semiclassical correction terms to the moment of inertia originating from the orbital motion and the spin degrees of freedom. Here we would like to examine the asymptotic behaviour of these terms in the hightemperature or low-density limit. For the sake of simplicity we shall restrict ourselves to only one kind of particles by considering a system of isoscalar nuclear matter (ρn = ρp = ρ/2). In thermodynamics of finite Fermi systems the high-temperature or low-density limit corresponds to the situation [66]  for which the average inter-particle distance is much larger than the thermal wavelength λ = 2π h¯ 2 /mT . One therefore expects quantum effects to be negligible. It can be mathematically shown that in the high-temperature or low-density limit the parameter η of the present formalism becomes negative (T → ∞ implies η → −∞). Consequently one can then use the following series expansion of the Fermi integrals Jµ (η) (see Refs. [34,49]) Jµ (η) = (µ + 1)

∞
(−1)k−1 k=1

k µ+1

ekη ,

η<0

(127)

which then leads in the asymptotic limit to the expression √ π η e , η 0. J1/2 (η) = (128) 2 For a pedagogical presentation it could be preferable, in order to understand exactly the hightemperature limit in the low-density case, to measure the temperature T relative to a Fermi temperature TF introduced below. Following Stoner [49] we start from the well-known relation between the density ρ and the Fermi energy εF of some fictitious free-particle system (including here a variable effective-mass form factor)   2m 3/2 3/2 1 ρ= εF . (129) 3π 2 h¯ 2 f If we define the Fermi temperature TF , which is then a function of the density, by exactly this energy TF = εF as usually done in quantum thermodynamics, and since the density is given by ρ = one then deduces   2 TF 3/2 J1/2 (η) = , 3 T

(130) A∗T J1/2 (η) (131)

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where we have used the definition of A∗T , Eq. (48). The high-temperature limit then corresponds to T TF . Using now the result of Eq. (128) we conclude that  3/2 TF 4 eη = √ (132) 3 π T and consequently     T 4 3 η = n √ − n 2 TF 3 π

(133)

which is already negative around T = TF and a fortiori for large temperatures. Putting the result of Eq. (128) into Eq. (101) one deduces that A∗T J−1/2 (η) = 2ρ,

η 0, T TF .

(134)

We thus conclude that in the absence of any effective mass the orbital correction to the moment of inertia in Eq. (119) would then behave like 1/T . It may be noted at this point that the Wigner–Kirkwood expansion in the presence of a vector-potential coupling was first developed by Jennings and Bhaduri in Ref. [67], where the semiclassical corrections to the diamagnetic susceptibility of an electron gas at zero as well as in the limiting case of high temperatures were obtained (up to order h¯ 4 ), in the absence of variable effective-mass and spin–orbit terms. To study the high-temperature limit of the paramagnetic correction to the moment of inertia, we consider the simple case of isoscalar nuclear matter where the spin susceptibility (χn = χp = χq ) solution of the system of Eqs. (126) is simply determined by   A∗T J−1/2 (η)

γ 1+ 2B10 + B11 + ρ (2B12 + B13 ) χq T   A∗T J−1/2 (η) 1 3 mB9 − = ρ . T 4 2 h¯ 2

(135)

In the high-temperature limit the second term of the l.h.s. of Eq. (135) can be neglected, according to Eq. (134), as compared to 1, which then yields the approximate form   2ρ 1 3 mB9 χq = ρ . (136) − T 4 2 h¯ 2 Inserting into the expression of the moment of inertia, Eq. (119), one notices that this paramagnetic term, too, behaves like 1/T . To conclude, we have shown that both the orbital (only for a unit effective-mass form factor) and the spin second-order semiclassical corrections to the moment of inertia behave for T TF as 1/T . Such an asymptotic behaviour is known from the description of magnetism of free electron systems. In fact it has been shown (see, e.g., Ref. [68]) that the diamagnetic susceptibility per unit volume behaves at high temperature (T TF ) as 1/T . Also it is well established that the paramagnetic susceptibility per unit volume at T TF follows Curie’s 1/T law (see, e.g., Ref. [69]). The above comparison might deepen and extend to finite temperatures our understanding of the analogy, known at T = 0, between the considered system of rotating fermions and the one of charged particles subject to a constant magnetic field.

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8. Conclusions We have given above a systematic derivation of the semiclassical Extended Thomas–Fermi (ETF) method (at order h¯ 2 ) for a N fermion system at finite temperature subject to a collective velocity field. We start from a Hamiltonian breaking time reversal symmetry which contains a linear coupling of classical collective vector fields to the linear momentum p of the particles. Spin degrees of freedom have also been included. Our approach is thus very general and applicable to different physical problems, such as, e.g., spin 1/2 particles in a homogeneous magnetic field, even if special emphasis has been given here to the case of rotating nuclei subject to some thermal excitation as they are obtained, e.g., as a result of heavy-ion collisions. The ETF approach allows to derive analytical functional expressions for different local quantities, such as the kinetic energy density τ already present in time-reversal symmetric systems, but also quantities as the current density j or the spin vector density ρ generated by the timereversal breaking part of the Hartree–Fock Hamiltonian. The latter allows, e.g., for the definition of a spin susceptibility which has been introduced to quantify the alignment of the spin vector density along the collective velocity. These local densities then determine a certain number of form factors, such as the mean-field potential, the cranking and the spin fields. We have also studied the low and high temperature limits of the different quantities determined in this way. Since all these quantities are, in the ETF method, functionals of the local density ρ we are able in the framework of Skyrme type effective interactions to perform selfconsistent density variational calculations in the spirit of density functional theory. This study constitutes a generalisation of work performed previously at finite temperature, but without any time-reversal symmetry breaking field [34], as well as in the presence of such fields but at zero temperature [35]. Among the integrated quantities which we can investigate in our approach, we have particularly studied the moment of inertia J . We have been able to determine the contributions of J originating from the different orders of our semiclassical expansion together with their temperature dependence. We thus find (as already noticed in [35]) that at the Thomas–Fermi level one recovers the rigid-body (RB) moment of inertia. Corrections to it are of diamagnetic type for the orbital motion and of paramagnetic type for those originating from the spin degrees of freedom. We find that as could be expected the RB contribution increases quadratically with the temperature T whereas the semiclassical (orbital and spin) corrections decrease with a well-known 1/T behaviour known as the Curie’s law. Our study is, in a way, a little disappointing in the nuclear case since, at least for integrated quantities such as the moment of inertia, the semiclassical corrections turn out to be rather small and, in addition, to compensate each other to a large extent in that case. It may be that for other applications, like for magnetic systems the effect might be more pronounced, in particular since for the magnetic case the Thomas–Fermi like term is zero. Thus, one is left with the semiclassical second-order terms which give rise to Landau diamagnetic and Pauli paramagnetic succeptibilities. In the framework of nuclear physics, an interesting application of the method which we have developed here would be to study fission barriers as function of both nuclear temperature and the angular velocity since it is well known that the nuclear stability decreases (i.e., the fissility increases when raising both nuclear temperature and angular momentum (or velocity).

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Acknowledgements The authors would like to thank Igor Mikhailov and David Samsoen for many helpful discussions in particular concerning vortical motion. One of us (K.B.) gratefully acknowledges the hospitality granted to him during many extended stays both at the CENBG in Bordeaux and the Institut de Recherches Subatomiques of Strasbourg. This work has been partially funded through the support of the CNRS-DEF collaboration agreement #17909 which is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

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