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Microcanonical averages and the validity of statistical treatments
.-3 FKI
27 July 1995
22%
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 35.5 (1995) 15-20
Microcanonical
averages and the validity of statistical treatments R. Rossignoli ‘, J.L. Egido
Departamento de Fisica Tedrica, Universidad Autdnoma de Madrid, 28049 Madrid, Spain Received 21 April 1995; revised manuscript Editor: C. Mahaux
received 8 June 1995
Abstract Microcanonical expectation values and their evaluation in the saddle point approximation are examined. Results are shown for a heavy nucleus in an independent particle model and in a double well model illustrating shape coexistence, where the exact microcanonical level density and average value of the quadmpole moment is compared with the grand canonical and canonical results.
Thermal treatments have been applied quite often recently to the theoretical investigation of highly excited nuclei, such as those formed in heavy ion fusion reactions. In heavy nuclei, the enormous number of many-body configurations accessible already at excitation energies of 5 MeV, seems to justify the application of statistical methods, even within effective models based on a restricted configuration space for valence nucleons, like the renowned Baranger-Kumar model [ 11. For a review of statistical descriptions, see for instance Ref. [ 21. Thermal treatments are usually implemented in the grand canonical (GC) ensemble, where calculations become straightforward in the finite temperature mean field approximation [ 3,4] or in higher order treatments based on an expansion over uncorrelated densities [ 591. Projection methods [ 10,111 are also available for performing thermal calculations in canonical (fixed particle number) ensembles [9,1 l-131 and in ensembles with fixed spin [ lo]. Nevertheless, heavy nuclei are still small quantum systems, and in principle the
1Permanent Address: Departamento cional de La Plata, Argentina.
de Ffsica, Universidad
ideal description should be based on the microcanonical ensemble, i.e., on an ensemble with fixed values of the excitation energy and other many-body observables like particle number and spin. The aim of this work is to examine the validity of GC and canonical statistical treatments and expectation values in heavy nuclei, and to extend the application of the saddle point approximation, which is usually employed only for extracting the energy level density from the partition function, to the calculation of microcanonical expectation values. An explicit comparison with exact microcanonical averages will be made within the Baranger-Kumar configuration space (two major shells for valence protons and neutrons). An additional illustrative example utilizing a Fermi gas level density is also shown. Let us consider a system described by a Hamiltonian l?. The GC partition function is Z(&CX) =Trexp(-p&+&j)
Na-
0370.2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00715-6
where p is the inverse
of the temperature
T, fi is
16
R. Rossignoli, J.L. Egido /Physics
the particle number operator, (Y = p,z, with ,u the chemical potential, and p(E, N) = Tra(fi - E)Sa,N is the many-body energy density of states, i.e., the microcanonical partition function for energy E and particle number N. We shall assume that [A, A] = 0. The GC expectation value of an observable & is (&)@ =zQ(k=')/Z(b'),
(2)
where ZQ(CX,/~) =Trexp(-PA+&)& =
p ( E, N) Q ( E, N) eePEfaNdE, SC N (3)
with Q(E,N)
= Tr[S(&
- E)6i,,&]/p(E,N)
Letters B 355 (1995) 15-20 N_
alnZU%a> = (@pa, aa
which determine the stationary point of the upper integrand in (4). The numerator in (5) is just es, with S the GC entropy, whereas A is the CC covariance matrix of the operators -8, fl. Expression (5) constitutes in practice an excellent approximation to the exact level density averaged over a not too small energy bin, except for very small excitation energies. If PQ (E, N) also increases sharply with E and N, the same procedure can be applied for obtaining PQ (E, N) from ZQ (p, a), provided Q (E, N) is sufficiently smooth and does not change sign with E. For Q( E, N) bounded, the last requirement can be fulfilled by a suitable displacement $ --t & + Qo. Thus,
the
microcanonical average of &. In a small quantum system like a finite nucleus, the quantities actually desired are p (E, N) and Q (E, N) rather than Z(/3, a) and ($)pU, but the direct evaluation of these quantities is in general difficult. However, they can in principle be exactly recovered from the inverse Laplace-Fourier transforms of Z( p, CX) and
ZQ
PQ(J%N) M
(p’, a’)
eP’E-a’N (7)
(2%-)k/2Det[AQ(/?‘,a’)]1/2’
where ( AQ )ij = a2 In with p’, LY’determined from E, N through the new equations zQ/&$qj,
E__dlnzQ(P’9d)
aPf
= (A&)p,,&!)pw
3
ZQ(&a),
(P+is,a++) J(;Q)
co 27r
(
p”,
>
(E,N)=&
1
-000 xe(P+is)E-(a+i~)Nd~ds,
(4)
where PQ (E, N) 3 p( E, N) Q( E, N) and where p and LYare here arbitrary constants. If, as usually, the energy level density p(E, N) increases sharply with E and N, the inverse integral for p( E, N) can be quite accurately evaluated in the saddle point approximation, in which the logarithm of the integrand is expanded up to second order in p and LYaround its stationary point. This leads to the wellknown result [ 141
p(E,N)
=
Z(/3,cx)ePE-nN (2r)k/2Det[A(/3,
a)] li2’
(5)
where Aij = d2 In Z/aviavj, with 7 = (/3, a), k = 2 is the number of variables in 7, and p, LYare determined from E, N through the usual thermodynamic equations E=_alnZ(P,ff) a4
= (&%YY
Hence, it is to be remarked that for a given value of E and N, a different temperature and chemical potential now applies in (7). To first order in Sr] = 7’ - 7, Eq. (8) gives ST = A;’ (7) 60, where
Eqs. (7) and (5) yield
' {Det[AQ(P', 4
1/Det[A(P. ay>1}1/2’ (9)
In a standard GC treatment, Q (E, N) is identified with the thermal average (2), with (,& a) obtained from (E, N) by the standard relations (6)) which is actually an average for fixed temperature and chemical potential. Eq. (9) gives the average for fixed E and N, and
R. Rossignoli, J.L. Egido/Physics
includes the curvature effects, represented by the determinants. To first order in 87, the numerator in (9) becomes zQ(p,a)e--so+. For many component systems, or in general, for level densities and averages with fixed quantum numbers Oi corresponding to observables & one should replace UYVby Ci CuiOi and (p, a) by (p, {LYE})in the previous expressions. In order to test the validity of the saddle point approximation for expectation values, we consider frrst a demonstrative example based on a level density simi-
Letters B 355 (1995) 15-20
17
7
6 5 @4 a
Exact C
3 2 1 0
10
20
lar to that of a Fermi gas, p(E) c( e2m/E’5/4, with E’ = E + Eo, a the level density parameter and E the excitation energy. We consider a canonical ensemble, i.e., 2 = sooo p(E) e-pEdE, and examine the recovery of Q(E) from Z(/3) and Z,(p) (cu is to beexcluded now from Eqs. (5)-(g)). We take a = 15 MeV-‘, EO = 1 MeV. We consider first the “observables”
Ql = god@ Qs=qo[l
-tanh(E/A)l,
- +nh[(E-&)/Al)
and an oscillatory QS =qo(l+
30
40
30
40
E (MW
Q2 = go/m,
with h = 10 MeV The choice of go is irrelevant. It is seen in Fig. 1 that the canonical average (&)p = ZQ(P)/Z(P), with E = -8 In Z/J& is always smoother than Q(E), although it provides a good estimate, whereas the improved expression (9) directly coincides with the exact Q(E) in the scale of the figure, except for very small E. In the case of Q2 and Qs, a displacement Qo can be applied for high E in order to avoid problems with the denominator of Eq. (8). For the level density, the saddle point approximation is very accurate. The error is less than 1% for E > 1.5 MeV. We depict in the lower figure a similar comparison for a smoothed step function Qd=qo(l
----
function
;cosE/A),
with El = 10, A = 5 MeV. Changes in the curvature of Q(E) now occur. Less accurate results are now obtained, particularly for the oscillatory observable, showing that the present methods are unable to reproduce sudden changes and oscillatory behavior in
0'
10
20
1
E (MW
Fig. 1. Microcanonical (Exact) and canonical (C) averages of the functions (see text) Q1 (a), Q2 (b) and Q3 (c) in the upper figure and Q4 (a) and Q5 (b) in the lower figure, employing a Fermi gas level density. Cfsp indicates the saddle point approximation (9) (in the canonical ensemble), almost undistinguishable from the exact result in tbe upper figure. Q(E) . Nonetheless, Eq. (9) still provides an improvement over the canonical average in both cases, indicating that a deviation between the canonical and microcanonical average is taking place. As a realistic example, we consider a calculation in the Baranger-Kumar configuration space for 164Er. We use first a deformed single particle (sp) Hamiltonian & = &a - fiw/3&20, where & is the spherical sp part employed in Ref. [ 1] and &2a is the quadrupole operator (scaled as in Ref. [l] ), with & the ground state deformation. In this case we can calculate microcanonical averages exactly by summing particle-hole (ph) excitations. In the GC ensemble,
Z(/3,aP,a,)
=Det(l
+e-gh+apnrp+~nNn),
(10)
where h and Np,” are the sp matrices representing h and the proton and neutron number operators. Wick’s theorem allows to calculate (&)P~~~,~ for any many-
18
R. Rossignoli, J.L. Egido/Physics Letters B 355 (199.5) IS-20
upto5ph
----
GC
2
4
6
8
10
-
12
14
E (MeV)
10
20
30
40
E (MeV) Fig. 2. Grand canonical (GC) and microcanonical average of the (scaled) quadrupole operator &2o in ‘a~, in a deformed sp Hamiltonian. In the upper figure, results using up to 1, 3 and 5 particle-hole excitations are shown, with an energy interval of 0.1 MeV. The lower figure depicts results in a larger energy scale (with the 5 ph result up to 15 MeV), where GCSsp denotes the approximation (9), almost undistinguishable from the GC average.
body operator $. Results are depicted in Fig. 2 for $ = &J. For the actual microcanonical calculation, we have set an energy bin A = 0.1 MeV, i.e., we have considered p(E) = ( l/A) s,“_‘,“//,”p( E)dE. For excitation energies E < 15 MeV, it is sufficient in this case to consider up to 5 ph excitations for each component, which involves up to 10 ph excitations for the combined p + n system (there are 2.8 x 10” states of this kind with E < 15 MeV, the dimension of the Baranger-Kumar many-body configuration space for this nucleus is 2.8 x 1043). Averages obtained using up to 3 and 5 ph excitations are practically coincident forE< 15MeV. It is first seen that for E > 5 MeV, no differences are found between the microcanonical and the GC average, whereas for E < 5 MeV, small amplitude fluctua-
tions appear in the microcanonical calculation, which are of course not reproduced by the statistical methods. No observable differences are found between the GC average and the saddle point expression (9) (now cy = ( CQ , a;, ) ) even for higher excitation energies, indicating that no mean deviation between the microcanonical and the GC average takes place. As a check, we have also chosen for & in Eq. (9) the operators fiP,n and &, in which case the particle number and the identity are reproduced with an error less than 0.1% and l%, respectively. An even better agreement is obtained for the level density with the expression (5)) the error being less than 12% for E > 5 MeV (see Ref. [ 131). Fluctuations are in this case smaller than for (&a), as will be seen in Fig. 4. We should also mention that no distinguishable differences are found between GC and canonical (fixed proton and neutron number) results, both for (&~a) and the level density, when plotted in terms of the corresponding excitation energy (for fixed temperature small differences arise [ 131) . The number projection can be done exactly by integration over cr in (4)) or in the saddle point approximation [ 131. In the present nucleus, and employing a twobody quadrupole-quadrupole interaction, the mean field equations exhibit a second oblate minimum at Pd z -0.25 (i.e., fid x 0.25, y = z-/3). Therefore, as a more stringent test for the statistical methods and in order to mimic some dynamical effects like shape coexistence, we examine a second example where an additional oblate well (described by a similar sp hamiltonian with /?d < 0, at an excitation energy Eo above the prolate minimum) is simultaneously considered. We also assume the second minimum and the corresponding ph excitations to be orthogonal to those of the prolate minimum and neglect tunnelling effects (we are considering only relatively small excitation energies). Hence, in this double well model the oblate configurations become accessible for E > EO which will be reflected in a decrease in (&a) (in the y = 0 convention). Since the chemical potentials are different for each well, we have performed a canonical calculation, settingZ(P) = Zr,(/3)+e-PEoZ0(p), whereZp.(P) are the canonical partition functions for each well (with energies measured from the respective minimum). We depict in Fig. 3 results for three different values of Eo, the case EO = 5 MeV being the most realistic one for
R. Rossigmoli, J.L. Egido/Physics
Letters B 35.5 (1995) 15-20
19
15 CL 9
-0.2 up to 5 ph -----
10
GC+sp
0 2
4
6
. . . . . .. . . . .
c+sp 8
10
12
5
14
-----
up to Sph
E (MeV) 0.3
qY>”
0
.\\,
*:
11 2
4
6
8
10
12
14
E (MeV) Fig. 4. Same details as Fig. 3 for the logarithm of the energy level density (in MeV-‘), for ~70 = 2.5 MeV. For each separate well the GC result is also depicted.
2
4
6
8
10
12
14
E (MeV) 0.3 ,,,
i
P
c
--
c+sp ---------.. up to 5 ph ----P
2
4
6
8
10
12
14
E (MeV) Fig. 3. Same details as Fig. 2, but adding a second oblate well at an excitation energy (in MeV) Eu = 5 (upper figure), Eu = 2.5 (center) and Eo = 0.5 (lower figure). Curves p (0) depict the averages for the prolate (oblate) well, whereas p+o for the combined system. The vertical line in the central figure indicates the prolate to oblate transition at E = 6.4 MeV of the mean field approximation.
the present and
employed
We have set again A = 0.1 MeV, a translation Qo = -(&20)~ in order to
nucleus.
apply Eq. (7). The oblate well possesses a larger density of states owing to the smaller average spacing between sp energies, and will strongly affect the value of Q(E) . Differences between the microcanonical and the canonical average now arise in the transitional region. The latter is smoother, being affected by the oblate components already for E < Eo, although results become again coincident for higher excitation energies, as seen
in the lowest figure. The approximation (9) improves the canonical average in all cases, except in the region just before the transition. Differences between the temperatures appearing in Eqs. (5) and (7) are of the order of 10% in the transitional region, decreasing towards 4% for E = 15 MeV. Large amplitude fluctuations in Q(E) are now observed for E around Eo in the microcanonical calculation, for EO = 0.5 and 2.5 MeV. In comparison, the fluctuations in Q(E) for each separate well are in the present scale practically undistinguishable. The saddle point approximation remains accurate for the level density (Fig. 4)) as this quantity is smooth. Fluctuations are much smaller and the error is less than 10% for E > 5 MeV. In a pure mean field approximation, the system is described by a single sp Hamiltonian, i.e., by a single well. For fixed values of E and N, the optimum well is that leading to the largest entropy. The mean field approach would thus exhibit in the present model two solutions (prolate and oblate) , with a sharp prolate to oblate transition at an energy EC where the entropies of the prolate and oblate solutions cross each other. These energies are EC = 0.63,6.4 and 20 for Eo = 0.5,2.5 and 5 MeV, respectively. Hence, although the canonical thermal average is smoother than the microcanonical average, the latter is still much smoother than the mean field result. In the correct picture both prolate and oblate components coexist for E > Eo. In conclusion, the evaluation of microcanonical expectation values in the saddle point approximation has been examined. The method provides a simple way
20
R. Rossignoli, J.L. Egido/Physics
to improve the thermal average (2), and will indicate if a mean deviation between the latter and the microcanonical average exists. Results obtained in a heavy nucleus within a double well model show that differences between the statistical (GC or canonical) and the microcanonical average may indeed arise in transitional regions for sensitive observables, although outside these regions differences will be typically small, except for the fluctuations visible in the exact microcanonical average at small excitation energies. For smooth quantities like the level density, canonical and GC results are seen to be quite reliable, except for small excitation energies, in contrast with the suggestion of Ref. [ 151. We remark that classical arguments about level densities do not necessarily hold in quantum systems with undistinguishable particles (in particular, the level density of A undistinguishable fermions in a given sp space is not the convolution of the level densities of subsystems with A’ and A -A’ fermions in that space, even in a noninteracting picture) . A more realistic study along these lines of the reliability of statistical treatments indicating a strong smoothing of the phase transitions found in thermal mean field approximations is under investigation. This work was supported in part by DGICyT, Spain, under project PB9 l-0006 (J.L.E. ) and a grant for Estancias Temporales de Cientificos y Tecnologos Extranjeros (R.R.) .R.R. is a fellow of CICPBA, of Argentina, and acknowledges a grant from Fundacion Antorchas.
Letters B 355 (1995) 15-20
References [ 11 M. Baranger and K. Knmar, Nncl. Phys. A 110 (1968) 490; K. Knmar and M. Baranger, Nncl. Phys. A 110 (1968) 529. [2] J.L. Egido and F’.Ring, J. Phys. G 19 (1993) 1. [3] A.L. Goodman, Nncl. Phys. A 352 (1981) 30. [4] A.L. Goodman, Phys. Rev. C 29 (1984) 1887; C 37 (1988) 2162. [5] J. Hubbard, Phys. Rev. I.&t. 3 (1959) 77; R.L. Stratonovich, Dokl. Akad. Nank. SSSR 115 (1957) 1097 [Soviet Phys. Dokl. 2 (1958) 4581. [6] Y. Alhassid and J. Zingman, Phys. Rev. C 30 (1984) 684. [7] P. Arve, G.F. Bertsch, B. Lanritzen and G. Pnddn, Ann. Phys. (N.Y.) 183 (1988) 309. [ 81 Y. Alhassid and B. Bush, Nncl. Phys. A 549 (1992) 43; A 565 (1993) 399. [9] G.H. Lang, C.W. Johnson, S.E. Koonin and W.E. Ormand, Phys. Rev. C 48 (1993) 1518; W.E. Ormand, D.J. Dean, C.W. Johnson, G.H. Lang and S.E. Koonin, Phys. Rev. C 49 (1994) 1422. [lo] R. Rossignoli and P. Ring, Ann. Phys. (N.Y.) 235 (1994) 350; R. Rossignoli, A. Ansari and P. Ring, Phys. Rev. Lett. 70 (1993) 1061. [ 1I] C. Essebag and J.L. Egido, Nacl. Phys. A 552 (1993) 205. 1121 R. Rossignoli, P. Ring and N. Dang, Phys. L&t. B 297 (1992) 9. [ 131 R. Rossignoli, Pbys. Rev. C 51 ( 1995) 1772. 1141 A. Bohr and B. Mottelson, Nuclear structure, vol. 1 (Benjamin, Reading, MA, 1969). [15] D.H.E. Gross and R. Heck, Phys. Lett. B 318 (1993) 405.
27 July 1995
22%
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 35.5 (1995) 15-20
Microcanonical
averages and the validity of statistical treatments R. Rossignoli ‘, J.L. Egido
Departamento de Fisica Tedrica, Universidad Autdnoma de Madrid, 28049 Madrid, Spain Received 21 April 1995; revised manuscript Editor: C. Mahaux
received 8 June 1995
Abstract Microcanonical expectation values and their evaluation in the saddle point approximation are examined. Results are shown for a heavy nucleus in an independent particle model and in a double well model illustrating shape coexistence, where the exact microcanonical level density and average value of the quadmpole moment is compared with the grand canonical and canonical results.
Thermal treatments have been applied quite often recently to the theoretical investigation of highly excited nuclei, such as those formed in heavy ion fusion reactions. In heavy nuclei, the enormous number of many-body configurations accessible already at excitation energies of 5 MeV, seems to justify the application of statistical methods, even within effective models based on a restricted configuration space for valence nucleons, like the renowned Baranger-Kumar model [ 11. For a review of statistical descriptions, see for instance Ref. [ 21. Thermal treatments are usually implemented in the grand canonical (GC) ensemble, where calculations become straightforward in the finite temperature mean field approximation [ 3,4] or in higher order treatments based on an expansion over uncorrelated densities [ 591. Projection methods [ 10,111 are also available for performing thermal calculations in canonical (fixed particle number) ensembles [9,1 l-131 and in ensembles with fixed spin [ lo]. Nevertheless, heavy nuclei are still small quantum systems, and in principle the
1Permanent Address: Departamento cional de La Plata, Argentina.
de Ffsica, Universidad
ideal description should be based on the microcanonical ensemble, i.e., on an ensemble with fixed values of the excitation energy and other many-body observables like particle number and spin. The aim of this work is to examine the validity of GC and canonical statistical treatments and expectation values in heavy nuclei, and to extend the application of the saddle point approximation, which is usually employed only for extracting the energy level density from the partition function, to the calculation of microcanonical expectation values. An explicit comparison with exact microcanonical averages will be made within the Baranger-Kumar configuration space (two major shells for valence protons and neutrons). An additional illustrative example utilizing a Fermi gas level density is also shown. Let us consider a system described by a Hamiltonian l?. The GC partition function is Z(&CX) =Trexp(-p&+&j)
Na-
0370.2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00715-6
where p is the inverse
of the temperature
T, fi is
16
R. Rossignoli, J.L. Egido /Physics
the particle number operator, (Y = p,z, with ,u the chemical potential, and p(E, N) = Tra(fi - E)Sa,N is the many-body energy density of states, i.e., the microcanonical partition function for energy E and particle number N. We shall assume that [A, A] = 0. The GC expectation value of an observable & is (&)@ =zQ(k=')/Z(b'),
(2)
where ZQ(CX,/~) =Trexp(-PA+&)& =
p ( E, N) Q ( E, N) eePEfaNdE, SC N (3)
with Q(E,N)
= Tr[S(&
- E)6i,,&]/p(E,N)
Letters B 355 (1995) 15-20 N_
alnZU%a> = (@pa, aa
which determine the stationary point of the upper integrand in (4). The numerator in (5) is just es, with S the GC entropy, whereas A is the CC covariance matrix of the operators -8, fl. Expression (5) constitutes in practice an excellent approximation to the exact level density averaged over a not too small energy bin, except for very small excitation energies. If PQ (E, N) also increases sharply with E and N, the same procedure can be applied for obtaining PQ (E, N) from ZQ (p, a), provided Q (E, N) is sufficiently smooth and does not change sign with E. For Q( E, N) bounded, the last requirement can be fulfilled by a suitable displacement $ --t & + Qo. Thus,
the
microcanonical average of &. In a small quantum system like a finite nucleus, the quantities actually desired are p (E, N) and Q (E, N) rather than Z(/3, a) and ($)pU, but the direct evaluation of these quantities is in general difficult. However, they can in principle be exactly recovered from the inverse Laplace-Fourier transforms of Z( p, CX) and
ZQ
PQ(J%N) M
(p’, a’)
eP’E-a’N (7)
(2%-)k/2Det[AQ(/?‘,a’)]1/2’
where ( AQ )ij = a2 In with p’, LY’determined from E, N through the new equations zQ/&$qj,
E__dlnzQ(P’9d)
aPf
= (A&)p,,&!)pw
3
ZQ(&a),
(P+is,a++) J(;Q)
co 27r
(
p”,
>
(E,N)=&
1
-000 xe(P+is)E-(a+i~)Nd~ds,
(4)
where PQ (E, N) 3 p( E, N) Q( E, N) and where p and LYare here arbitrary constants. If, as usually, the energy level density p(E, N) increases sharply with E and N, the inverse integral for p( E, N) can be quite accurately evaluated in the saddle point approximation, in which the logarithm of the integrand is expanded up to second order in p and LYaround its stationary point. This leads to the wellknown result [ 141
p(E,N)
=
Z(/3,cx)ePE-nN (2r)k/2Det[A(/3,
a)] li2’
(5)
where Aij = d2 In Z/aviavj, with 7 = (/3, a), k = 2 is the number of variables in 7, and p, LYare determined from E, N through the usual thermodynamic equations E=_alnZ(P,ff) a4
= (&%YY
Hence, it is to be remarked that for a given value of E and N, a different temperature and chemical potential now applies in (7). To first order in Sr] = 7’ - 7, Eq. (8) gives ST = A;’ (7) 60, where
Eqs. (7) and (5) yield
' {Det[AQ(P', 4
1/Det[A(P. ay>1}1/2’ (9)
In a standard GC treatment, Q (E, N) is identified with the thermal average (2), with (,& a) obtained from (E, N) by the standard relations (6)) which is actually an average for fixed temperature and chemical potential. Eq. (9) gives the average for fixed E and N, and
R. Rossignoli, J.L. Egido/Physics
includes the curvature effects, represented by the determinants. To first order in 87, the numerator in (9) becomes zQ(p,a)e--so+. For many component systems, or in general, for level densities and averages with fixed quantum numbers Oi corresponding to observables & one should replace UYVby Ci CuiOi and (p, a) by (p, {LYE})in the previous expressions. In order to test the validity of the saddle point approximation for expectation values, we consider frrst a demonstrative example based on a level density simi-
Letters B 355 (1995) 15-20
17
7
6 5 @4 a
Exact C
3 2 1 0
10
20
lar to that of a Fermi gas, p(E) c( e2m/E’5/4, with E’ = E + Eo, a the level density parameter and E the excitation energy. We consider a canonical ensemble, i.e., 2 = sooo p(E) e-pEdE, and examine the recovery of Q(E) from Z(/3) and Z,(p) (cu is to beexcluded now from Eqs. (5)-(g)). We take a = 15 MeV-‘, EO = 1 MeV. We consider first the “observables”
Ql = god@ Qs=qo[l
-tanh(E/A)l,
- +nh[(E-&)/Al)
and an oscillatory QS =qo(l+
30
40
30
40
E (MW
Q2 = go/m,
with h = 10 MeV The choice of go is irrelevant. It is seen in Fig. 1 that the canonical average (&)p = ZQ(P)/Z(P), with E = -8 In Z/J& is always smoother than Q(E), although it provides a good estimate, whereas the improved expression (9) directly coincides with the exact Q(E) in the scale of the figure, except for very small E. In the case of Q2 and Qs, a displacement Qo can be applied for high E in order to avoid problems with the denominator of Eq. (8). For the level density, the saddle point approximation is very accurate. The error is less than 1% for E > 1.5 MeV. We depict in the lower figure a similar comparison for a smoothed step function Qd=qo(l
----
function
;cosE/A),
with El = 10, A = 5 MeV. Changes in the curvature of Q(E) now occur. Less accurate results are now obtained, particularly for the oscillatory observable, showing that the present methods are unable to reproduce sudden changes and oscillatory behavior in
0'
10
20
1
E (MW
Fig. 1. Microcanonical (Exact) and canonical (C) averages of the functions (see text) Q1 (a), Q2 (b) and Q3 (c) in the upper figure and Q4 (a) and Q5 (b) in the lower figure, employing a Fermi gas level density. Cfsp indicates the saddle point approximation (9) (in the canonical ensemble), almost undistinguishable from the exact result in tbe upper figure. Q(E) . Nonetheless, Eq. (9) still provides an improvement over the canonical average in both cases, indicating that a deviation between the canonical and microcanonical average is taking place. As a realistic example, we consider a calculation in the Baranger-Kumar configuration space for 164Er. We use first a deformed single particle (sp) Hamiltonian & = &a - fiw/3&20, where & is the spherical sp part employed in Ref. [ 1] and &2a is the quadrupole operator (scaled as in Ref. [l] ), with & the ground state deformation. In this case we can calculate microcanonical averages exactly by summing particle-hole (ph) excitations. In the GC ensemble,
Z(/3,aP,a,)
=Det(l
+e-gh+apnrp+~nNn),
(10)
where h and Np,” are the sp matrices representing h and the proton and neutron number operators. Wick’s theorem allows to calculate (&)P~~~,~ for any many-
18
R. Rossignoli, J.L. Egido/Physics Letters B 355 (199.5) IS-20
upto5ph
----
GC
2
4
6
8
10
-
12
14
E (MeV)
10
20
30
40
E (MeV) Fig. 2. Grand canonical (GC) and microcanonical average of the (scaled) quadrupole operator &2o in ‘a~, in a deformed sp Hamiltonian. In the upper figure, results using up to 1, 3 and 5 particle-hole excitations are shown, with an energy interval of 0.1 MeV. The lower figure depicts results in a larger energy scale (with the 5 ph result up to 15 MeV), where GCSsp denotes the approximation (9), almost undistinguishable from the GC average.
body operator $. Results are depicted in Fig. 2 for $ = &J. For the actual microcanonical calculation, we have set an energy bin A = 0.1 MeV, i.e., we have considered p(E) = ( l/A) s,“_‘,“//,”p( E)dE. For excitation energies E < 15 MeV, it is sufficient in this case to consider up to 5 ph excitations for each component, which involves up to 10 ph excitations for the combined p + n system (there are 2.8 x 10” states of this kind with E < 15 MeV, the dimension of the Baranger-Kumar many-body configuration space for this nucleus is 2.8 x 1043). Averages obtained using up to 3 and 5 ph excitations are practically coincident forE< 15MeV. It is first seen that for E > 5 MeV, no differences are found between the microcanonical and the GC average, whereas for E < 5 MeV, small amplitude fluctua-
tions appear in the microcanonical calculation, which are of course not reproduced by the statistical methods. No observable differences are found between the GC average and the saddle point expression (9) (now cy = ( CQ , a;, ) ) even for higher excitation energies, indicating that no mean deviation between the microcanonical and the GC average takes place. As a check, we have also chosen for & in Eq. (9) the operators fiP,n and &, in which case the particle number and the identity are reproduced with an error less than 0.1% and l%, respectively. An even better agreement is obtained for the level density with the expression (5)) the error being less than 12% for E > 5 MeV (see Ref. [ 131). Fluctuations are in this case smaller than for (&a), as will be seen in Fig. 4. We should also mention that no distinguishable differences are found between GC and canonical (fixed proton and neutron number) results, both for (&~a) and the level density, when plotted in terms of the corresponding excitation energy (for fixed temperature small differences arise [ 131) . The number projection can be done exactly by integration over cr in (4)) or in the saddle point approximation [ 131. In the present nucleus, and employing a twobody quadrupole-quadrupole interaction, the mean field equations exhibit a second oblate minimum at Pd z -0.25 (i.e., fid x 0.25, y = z-/3). Therefore, as a more stringent test for the statistical methods and in order to mimic some dynamical effects like shape coexistence, we examine a second example where an additional oblate well (described by a similar sp hamiltonian with /?d < 0, at an excitation energy Eo above the prolate minimum) is simultaneously considered. We also assume the second minimum and the corresponding ph excitations to be orthogonal to those of the prolate minimum and neglect tunnelling effects (we are considering only relatively small excitation energies). Hence, in this double well model the oblate configurations become accessible for E > EO which will be reflected in a decrease in (&a) (in the y = 0 convention). Since the chemical potentials are different for each well, we have performed a canonical calculation, settingZ(P) = Zr,(/3)+e-PEoZ0(p), whereZp.(P) are the canonical partition functions for each well (with energies measured from the respective minimum). We depict in Fig. 3 results for three different values of Eo, the case EO = 5 MeV being the most realistic one for
R. Rossigmoli, J.L. Egido/Physics
Letters B 35.5 (1995) 15-20
19
15 CL 9
-0.2 up to 5 ph -----
10
GC+sp
0 2
4
6
. . . . . .. . . . .
c+sp 8
10
12
5
14
-----
up to Sph
E (MeV) 0.3
qY>”
0
.\\,
*:
11 2
4
6
8
10
12
14
E (MeV) Fig. 4. Same details as Fig. 3 for the logarithm of the energy level density (in MeV-‘), for ~70 = 2.5 MeV. For each separate well the GC result is also depicted.
2
4
6
8
10
12
14
E (MeV) 0.3 ,,,
i
P
c
--
c+sp ---------.. up to 5 ph ----P
2
4
6
8
10
12
14
E (MeV) Fig. 3. Same details as Fig. 2, but adding a second oblate well at an excitation energy (in MeV) Eu = 5 (upper figure), Eu = 2.5 (center) and Eo = 0.5 (lower figure). Curves p (0) depict the averages for the prolate (oblate) well, whereas p+o for the combined system. The vertical line in the central figure indicates the prolate to oblate transition at E = 6.4 MeV of the mean field approximation.
the present and
employed
We have set again A = 0.1 MeV, a translation Qo = -(&20)~ in order to
nucleus.
apply Eq. (7). The oblate well possesses a larger density of states owing to the smaller average spacing between sp energies, and will strongly affect the value of Q(E) . Differences between the microcanonical and the canonical average now arise in the transitional region. The latter is smoother, being affected by the oblate components already for E < Eo, although results become again coincident for higher excitation energies, as seen
in the lowest figure. The approximation (9) improves the canonical average in all cases, except in the region just before the transition. Differences between the temperatures appearing in Eqs. (5) and (7) are of the order of 10% in the transitional region, decreasing towards 4% for E = 15 MeV. Large amplitude fluctuations in Q(E) are now observed for E around Eo in the microcanonical calculation, for EO = 0.5 and 2.5 MeV. In comparison, the fluctuations in Q(E) for each separate well are in the present scale practically undistinguishable. The saddle point approximation remains accurate for the level density (Fig. 4)) as this quantity is smooth. Fluctuations are much smaller and the error is less than 10% for E > 5 MeV. In a pure mean field approximation, the system is described by a single sp Hamiltonian, i.e., by a single well. For fixed values of E and N, the optimum well is that leading to the largest entropy. The mean field approach would thus exhibit in the present model two solutions (prolate and oblate) , with a sharp prolate to oblate transition at an energy EC where the entropies of the prolate and oblate solutions cross each other. These energies are EC = 0.63,6.4 and 20 for Eo = 0.5,2.5 and 5 MeV, respectively. Hence, although the canonical thermal average is smoother than the microcanonical average, the latter is still much smoother than the mean field result. In the correct picture both prolate and oblate components coexist for E > Eo. In conclusion, the evaluation of microcanonical expectation values in the saddle point approximation has been examined. The method provides a simple way
20
R. Rossignoli, J.L. Egido/Physics
to improve the thermal average (2), and will indicate if a mean deviation between the latter and the microcanonical average exists. Results obtained in a heavy nucleus within a double well model show that differences between the statistical (GC or canonical) and the microcanonical average may indeed arise in transitional regions for sensitive observables, although outside these regions differences will be typically small, except for the fluctuations visible in the exact microcanonical average at small excitation energies. For smooth quantities like the level density, canonical and GC results are seen to be quite reliable, except for small excitation energies, in contrast with the suggestion of Ref. [ 151. We remark that classical arguments about level densities do not necessarily hold in quantum systems with undistinguishable particles (in particular, the level density of A undistinguishable fermions in a given sp space is not the convolution of the level densities of subsystems with A’ and A -A’ fermions in that space, even in a noninteracting picture) . A more realistic study along these lines of the reliability of statistical treatments indicating a strong smoothing of the phase transitions found in thermal mean field approximations is under investigation. This work was supported in part by DGICyT, Spain, under project PB9 l-0006 (J.L.E. ) and a grant for Estancias Temporales de Cientificos y Tecnologos Extranjeros (R.R.) .R.R. is a fellow of CICPBA, of Argentina, and acknowledges a grant from Fundacion Antorchas.
Letters B 355 (1995) 15-20
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