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What is wrong with the Bethe-formula for the nuclear level-density? measurable differences between grandcanonical and the microcanonical treatments
PhysicsLetters B 318 (1993) 405-409 North-Holland
PHYSICS LETTERS B
What is wrong with the Bethe-formula for the nuclear level-density? Measurable differences between grandcanonical and the microcanonical treatments D.H.E. Gross a and R. H e c k a,b a Hahn-Meitner-lnstitut, Bereich Kern- und Strahlenphysik, Glienickerstr. I00, D-14109 Berlin, Germany b Fachbereich Physik der Freien Universit~it, Berlin, Germany Received 10 August 1993; revised manuscript received 17 September 1993 Editor: C. Mahaux The familiar Bethe formula is the leading term (pc volume) in an asymptotic leptodermous expansion of the nuclear level-density for very large neutron (N), proton (Z) number, and excitation energy E*. It is shown that for realistic finite nuclei it has a wrong dependence on mass and energy by which the internal phase-space for the split of a heavier nucleus into two similar pieces is reduced. This leads e.g. to a substantial and - measurable - suppression of statistical fragmentation into larger fragments of intermediate mass which contradicts to existing data of multifragmentation. Consequently one cannot extrapolate the level-density with the Bethe formula beyond the region of experimental control even if various higher leptodermous corrections are incorporated.
it is a stationary phase a p p r o x i m a t i o n ) . That is, one replaces the logarithm o f the integrand in
1. Introduction At low energies the nuclear level-density is well known. Various corrections to the standard Betheformula (the level-density parameter a is ~ A = N+Z) pA(E) -
it 1/2 e2Vfi-g 12ES/4 al/4 ,
× e -(u~N+uzz)/r e E/r dUN d g z d ( 1 / T )
0
(2)
by triple Laplace-back transform [ 1]. This is performed by the saddle-point method (strictly speaking
(3)
by its quadratic approximant, or Z (#~,/~z, T) by a Gaussian in #N, g z , 1/ T. Some few further conditions have to be fulfilled [ 1 ]: Conditions: (1) Replacing the single-particle level-density
g(e) = ~ " ~ 6 ( e - e ~ )
Z (#N, # z , T)
e - E / r e (uNN+uzz)/r d N d Z d E
c+ioo
c-ioo
(1)
are proposed to take curvature-, isospin-, deformations-, pairing-, and shell effects into account. These corrected forms are fitted to data o f neutron resonances near to the neutron-threshold. F o r m u l a ( 1 ) is obtained from the grandcanonical partition sum o f an ideal Fermi-gas:
x
pN, z ( E ) = Z ( N , Z , E )
(4)
by a smooth function in e, this amounts to the condition g (eF) T >> 1, and (2) neglect terms proportional to derivatives o f g (e), i.e. one approximates g(e) ~ g(eF).
0370-2693/93/$ 06.00 ~) 1993-Elsevier Science Publishers B.V. All rights reserved
(5) 405
Volume 318, number 3
PHYSICS LETTERS B
This limits the excitation energy to E* << ~FA 1/3 [ 1 ]. Then the integral of eq. (3) can be done and yields the Bethe formula eq. (1). This is a typical asymptotic approximation valid for large N, Z,E*, where the fluctuations can be ignored relative to the mean value of N, Z, E*. In this limit the canonical and the microcanonical ensembles coincide. For a finite realistic nucleus e.g. A = 100 at a sharp excitation of e.g. 100 MeV this may not be a sufficient approximation and leads to problems as we will see.
9 December 1993 A = 1 0 0 , E = 1 0 0 MeV
it
. . . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
lzi
]
~oa ",,
/
v
0.6
l
~
0.4
;!
,,,,,
,,/
0.2 0,0
In this section we will discuss some simple constraints on its analytical form which any acceptable level-density formula for a realistic nucleus has to fulfill. The Bethe formula is derived from the grandcanonical Fermi-gas. The chemical potentials (Fermienergies) /tN,/~z are assumed to depend only on the average density 0A = 0N + OZ. The free-energy F ( N , Z , T) as well as also the level-density parameter a depend only on the total number of nucleons A = N + Z. Consequently, it is the leading volume term in a leptodermic expansion of the Fermi-gas. Corrections of higher order like isospin-, surface-, curvature-, deformation-, shell-, and pairing effects are usually added later. If this is so the folded leveldensity of two nuclei with Al and A2 E*
fold (E*) = f P.~I+A,
pAl(E* - e ) p a , ( e ) de
(6)
0
should also depend in its leading order on the total mass A = At + Az only. We will call this the necessary "microcanonical analyticity constraint" ( M A C ) . This is, however, not the case if the Bethe-formula (1) is taken for pa(E) in eq. (6) as can be seen in fig. (1), where the ratio R(A, AI,E*) of the folded level-density p~d+A2 to the level-density of the mother nucleus is plotted for different mass-splits A1,A2 = A - A I. In the case shown the symmetric split is suppressed by a factor 1/ 10 compared to an enhancement of extremely asymmetric splitting. (Here and in the following we use a = A/16 instead of the value a = A/8 which is more appropriate at very low excitations 406
....
0
2. Measurable consequences
' ....
10
I ....
20
= ....
30
i,
.
, ....
i ....
40 50 60 m a s s AI
, ....
70
, ....
B0
i ....
90
I00
Fig. I. The ratio R (A, AI, E* ) of the folded density of states of the daughter nuclei over the density of states of the mother when a nucleus with mass A = I00 at excitation energy E* -- 100 MeV is split into two pieces with mass A i, A - A I . The level-densities are taken in their Bethe-approximation ( I ) and it is assumed that the Q-value is identical to zero (dashed curve). The divergence of the Bethe-formula at small excitation energies has been removed. The full line gives the same quantity, for the density ofeq. (8) which has the correct analytical form demanded by constraint MAC. and which simulates some additional surface corrections and therefore is not adequate for our discussion here. Moreover, this value o f the level-density parameter a would even sharpen the problem with the Bethe formula discussed here. For a = A/8 the suppression of the symmetric splitting (see fig. ( 1 ) ) is even more than a factor 1/ 10. However, the actual value of a/A is irrelevant for the arguments presented here against the analytic dependency of the Bethe density (1) on
A,E.) This suppression has o f course direct consequences for the mass-distribution of statistical multifragrnenration. In fig. (2) we show the relative probability of multifragmentation events of m X e as a function of excitation energy for three different choices o f the fragment level-density: ( 1 ) MAC, a level-density (formula (8)) which fulfills M A C , (2) Bethe, a level-density according to the Betheformula ( 1 ), (3) Bethe • I 0, a form "corrected" by simply multiplying the Bethe formula ( 1 ) by a constant factor 10. (In this calculation the sampling of the excited states in the primary fragments was carded through in a dif-
Volume 318, number 3
PHYSICS LETTERS B
9 December 1993 131Xe ,before evaporation
13tXe*, Multi f r a g m e n t a t i o n 1.0
. . . .
,
. . . .
,
0.9 .~
0.8
. . . .
,
10.0
. . . .
.
.
.
.
- - Multifragm., formula(B) . . . . Multifragm.,Bethe • .... .., *
...
'
.
.
.
.
'
4
- - formula(8) - - - Bet.he
"""
..... Bethe*lO
0.7 ~
0.6
~
0.5
......""
I 5.0
......
.."
//
~
0.4 ...." f / -~
0.3
o.2 0.1 0.0
500
1000
1500
0.0
2000
0
500
Fig. 2. Relative yield of multifragmentation events of 13tXe. (at least three fragments with A >I 10) as function of the excitation energy for three different level-densities of the excited primary fragments. As can be clearly seen the Bethe formula ( 1) suppresses the multifragmentation. ferent way than described in refs. [2,3]: They were sampled proportional to the density o f excited states and the evaporation of secondary neutrons, protons, and a-s was calculated with the evaporation code Julian [4]. This will be published in a separate publication [5 ]. The resulting fragment distributions are close to the ones o f the old B e r l i n - M M M C model as described in refs. [2,3,6] where M A C was build in explicitly from the beginning.) Evidently the yield o f multifragmentation is suppressed if the Bethe-formula is taken for the internal level-density of primaries at freeze-out instead of the - in this respect - more correct formula (8) which takes account of M A C . The enhancement of the very asymmetric splitting can be seen in fig. (3) in an enhancement of the production of He-isotopes. The corresponding information is given by fig. (4), where the mass of the three largest fragments is given for the decay of 131Xe* as function of the excitation energy. Again we see that the Bethe formula ( 1 ) gives considerably larger heaviest fragments. If the suppression o f the split into more or less symmetric fragments may be "corrected" by simply multiplying the Bethe level-density with a factor 10, the three heaviest masses and also the multifragmentation yield (fig. 2) reproduce roughly the result which uses the formula (8) and obeys M A C . Only at excitation energies above ~ 600 MeV the simple correction ( B e t h e . 10)
1000
1500
Energy (MeV)
Energy (MeV)
Fig. 3. Number of He-isotopes produced in the fragmentation of t3]Xe as function of the excitation energy. The full line is the calculation using the level-density according to formula (8), the dashed line gives the multiplicity of He using the Bethe formula (1), and the dotted curve shows the result of a simulation using the "corrected" Bethe formula B e t h e • 10. 131Xe, three largest fragments Formula(8)-Bethe, a=A/16 .... . .... . .... . ....
130 120
--
110
•
formula
(8)
---- Bethe
""
ioo 90 80
"-"N,
70 r~.
80
"\" k
50
.~ 40
+N,,,
30
",.
20 10 0
--i--0
500
1000
1500
2000
Energy (MeV) Fig. 4. The three largest masses of fragments produced in statistical fragmentation of 131Xe as simulated by microcanonical Metropolis Monte Carlo ( M M M C ) vs. excitation energy. gives too much fragmentation (too small heaviest fragments). Here the heaviest fragments have a mass ~ 50 and the suppression factor o f R ( A = 50, Al, E* ) is only ~, 1/5. Thus multiplying the Bethe formula by 10 enhances the fragmentation too much at higher excitations and the fragments come out too small. Correspondingly, the production of Heisotopes is strongly enhanced further (see fig. (3)). 407
Volume 318, number 3
PHYSICS LETTERS B
9 December 1993
Folding densities 60
i
E * = / O 0 Meg, a = A / 1 6 i i i i
i
i
,-.5O
40
20 /0
~ : 5 0
A=IO0 - - Besse l, {formula(8)] • asympt.Bessel • Bethe*e8
A1.='~2
1'0 ' 2~0 ' 3'0 ' 4~0 ' 5 0 ' 6~0 ' 7~0 ' 8~0 ' 9~0 ' / 0 0
e{MeV} Fig. 5. The integrand of the folding integral eq. (6) leading to R (A, Al, E*) as a function of e for the split of a nucleus with mass A -. A1 = 2, 10,25,50. One may think the failure of the Bethe formula to satisfy M A C as being due to the asymptotic approximation (saddle-point approximation in the Laplace back-transform (3) ). This is certainly the real origin of the problem but is not so straight forward: To visualize the problem, we assume for simplicity that the canonical partition sum Z (A, T) is given by the following Fermi-gas formula which is valid provided the conditions listed above are satisfied: Z ( A , T ) = c ear .
(7)
The single Laplace-back transform of (7) can be done exactly and yields, as discussed by Krappe [7 ], Z(A,E)
= J(E +0) +
I~(2v/~),
(9)
Fig. (5) shows the integrand of the folding eq. (6) as a function of e for the total mass A = 100 split in four different ways: .41 = 2, 10, 25, 50. Apart from the divergence of the asymptotic form at small energies e which can easily be removed (regularized) the agreement with the exact result is excellent. I.e. the asymptotic (saddle-point) approximation for the single Laplace back-transform is not the origin of the failure of M A C . Evidently the discrepancy arises when several Laplace back-transforms are done subsequently. It is the grandcanonical two-particle origin of the Bethe formula (1) that leads to problems.
(8)
where Il (x) is the modified Bessel-function of first order. This result is, however, only correct for low excitation energies (see condition (2) above). In general for excitations where multifragmentations are important the assumption o f Z (A, T) oc eaT is not valid, cf. ref. [2,6] and the free energy is not quadratic in T, see e.g. the calculation of the caloric relation T (E) of ref. [2 ]. We can compare this with the above saddle-point approximation of the corresponding single Laplaceback transform which is of course the well known asymptotic form of the modified Bessel function: 408
al/4 e2v'~ . pcan,asympt " 27tl/2E3/4
3. Conclusion The constraints M A C on the dependence on E* and A of the correct level-density of a large, but finite nucleus with realistic size are quite severe. The leading, volume term of the leptodermous expansion of the level-density is different from the Bethe formula, which is correct only in the asymptotic limit of large mass and energy. This difference has noticeable - and measurable - consequences on the fragment size distribution of statistical mulfifragmentation.
Volume 318, number 3
PHYSICS LETTERS B
Acknowledgement We thank A. Ecker for his help in clarifying the combinatoric problem of the exact level-density of the equidistant model.
References [ 1 ] Aa. Bohr and B.R. Mottelson, Nuclear Structure (W.A. Benjamin, New York, 1969). [2] D.H.E. Gross, Rep. Prog. Phys. 53 (1990) 605. [3] D.H.E. Gross and K. Sneppen, submitted to Nucl. Phys. (1993).
9 December 1993
[4] H. Rossner, D. Hilscher, D.J. Hinde, B. Gebauer, M. Lehmann, M. Wilert and E. Mordhorst, Phys. Rev. C 40 (1989) 2629. [5] R. Heck, D.H.E. Gross and A.R. DeAngelis, in preparation (1993). [6] D.H.E. Gross, in Proc. Second European Biennial Workshop on Nuclear Physics (Megeve, France, March 29-April 2, 1993), ed. D. Guinet (World Scientific, Singapore, 1993). [7 ] H.J. Krappe, in: Dynamical Aspects of Nuclear Fission, eds. J. K_ristak and B.I. Pustylink (Institute of Physics, SASc, Bratislava 1991) p. 51.
409
PHYSICS LETTERS B
What is wrong with the Bethe-formula for the nuclear level-density? Measurable differences between grandcanonical and the microcanonical treatments D.H.E. Gross a and R. H e c k a,b a Hahn-Meitner-lnstitut, Bereich Kern- und Strahlenphysik, Glienickerstr. I00, D-14109 Berlin, Germany b Fachbereich Physik der Freien Universit~it, Berlin, Germany Received 10 August 1993; revised manuscript received 17 September 1993 Editor: C. Mahaux The familiar Bethe formula is the leading term (pc volume) in an asymptotic leptodermous expansion of the nuclear level-density for very large neutron (N), proton (Z) number, and excitation energy E*. It is shown that for realistic finite nuclei it has a wrong dependence on mass and energy by which the internal phase-space for the split of a heavier nucleus into two similar pieces is reduced. This leads e.g. to a substantial and - measurable - suppression of statistical fragmentation into larger fragments of intermediate mass which contradicts to existing data of multifragmentation. Consequently one cannot extrapolate the level-density with the Bethe formula beyond the region of experimental control even if various higher leptodermous corrections are incorporated.
it is a stationary phase a p p r o x i m a t i o n ) . That is, one replaces the logarithm o f the integrand in
1. Introduction At low energies the nuclear level-density is well known. Various corrections to the standard Betheformula (the level-density parameter a is ~ A = N+Z) pA(E) -
it 1/2 e2Vfi-g 12ES/4 al/4 ,
× e -(u~N+uzz)/r e E/r dUN d g z d ( 1 / T )
0
(2)
by triple Laplace-back transform [ 1]. This is performed by the saddle-point method (strictly speaking
(3)
by its quadratic approximant, or Z (#~,/~z, T) by a Gaussian in #N, g z , 1/ T. Some few further conditions have to be fulfilled [ 1 ]: Conditions: (1) Replacing the single-particle level-density
g(e) = ~ " ~ 6 ( e - e ~ )
Z (#N, # z , T)
e - E / r e (uNN+uzz)/r d N d Z d E
c+ioo
c-ioo
(1)
are proposed to take curvature-, isospin-, deformations-, pairing-, and shell effects into account. These corrected forms are fitted to data o f neutron resonances near to the neutron-threshold. F o r m u l a ( 1 ) is obtained from the grandcanonical partition sum o f an ideal Fermi-gas:
x
pN, z ( E ) = Z ( N , Z , E )
(4)
by a smooth function in e, this amounts to the condition g (eF) T >> 1, and (2) neglect terms proportional to derivatives o f g (e), i.e. one approximates g(e) ~ g(eF).
0370-2693/93/$ 06.00 ~) 1993-Elsevier Science Publishers B.V. All rights reserved
(5) 405
Volume 318, number 3
PHYSICS LETTERS B
This limits the excitation energy to E* << ~FA 1/3 [ 1 ]. Then the integral of eq. (3) can be done and yields the Bethe formula eq. (1). This is a typical asymptotic approximation valid for large N, Z,E*, where the fluctuations can be ignored relative to the mean value of N, Z, E*. In this limit the canonical and the microcanonical ensembles coincide. For a finite realistic nucleus e.g. A = 100 at a sharp excitation of e.g. 100 MeV this may not be a sufficient approximation and leads to problems as we will see.
9 December 1993 A = 1 0 0 , E = 1 0 0 MeV
it
. . . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
lzi
]
~oa ",,
/
v
0.6
l
~
0.4
;!
,,,,,
,,/
0.2 0,0
In this section we will discuss some simple constraints on its analytical form which any acceptable level-density formula for a realistic nucleus has to fulfill. The Bethe formula is derived from the grandcanonical Fermi-gas. The chemical potentials (Fermienergies) /tN,/~z are assumed to depend only on the average density 0A = 0N + OZ. The free-energy F ( N , Z , T) as well as also the level-density parameter a depend only on the total number of nucleons A = N + Z. Consequently, it is the leading volume term in a leptodermic expansion of the Fermi-gas. Corrections of higher order like isospin-, surface-, curvature-, deformation-, shell-, and pairing effects are usually added later. If this is so the folded leveldensity of two nuclei with Al and A2 E*
fold (E*) = f P.~I+A,
pAl(E* - e ) p a , ( e ) de
(6)
0
should also depend in its leading order on the total mass A = At + Az only. We will call this the necessary "microcanonical analyticity constraint" ( M A C ) . This is, however, not the case if the Bethe-formula (1) is taken for pa(E) in eq. (6) as can be seen in fig. (1), where the ratio R(A, AI,E*) of the folded level-density p~d+A2 to the level-density of the mother nucleus is plotted for different mass-splits A1,A2 = A - A I. In the case shown the symmetric split is suppressed by a factor 1/ 10 compared to an enhancement of extremely asymmetric splitting. (Here and in the following we use a = A/16 instead of the value a = A/8 which is more appropriate at very low excitations 406
....
0
2. Measurable consequences
' ....
10
I ....
20
= ....
30
i,
.
, ....
i ....
40 50 60 m a s s AI
, ....
70
, ....
B0
i ....
90
I00
Fig. I. The ratio R (A, AI, E* ) of the folded density of states of the daughter nuclei over the density of states of the mother when a nucleus with mass A = I00 at excitation energy E* -- 100 MeV is split into two pieces with mass A i, A - A I . The level-densities are taken in their Bethe-approximation ( I ) and it is assumed that the Q-value is identical to zero (dashed curve). The divergence of the Bethe-formula at small excitation energies has been removed. The full line gives the same quantity, for the density ofeq. (8) which has the correct analytical form demanded by constraint MAC. and which simulates some additional surface corrections and therefore is not adequate for our discussion here. Moreover, this value o f the level-density parameter a would even sharpen the problem with the Bethe formula discussed here. For a = A/8 the suppression of the symmetric splitting (see fig. ( 1 ) ) is even more than a factor 1/ 10. However, the actual value of a/A is irrelevant for the arguments presented here against the analytic dependency of the Bethe density (1) on
A,E.) This suppression has o f course direct consequences for the mass-distribution of statistical multifragrnenration. In fig. (2) we show the relative probability of multifragmentation events of m X e as a function of excitation energy for three different choices o f the fragment level-density: ( 1 ) MAC, a level-density (formula (8)) which fulfills M A C , (2) Bethe, a level-density according to the Betheformula ( 1 ), (3) Bethe • I 0, a form "corrected" by simply multiplying the Bethe formula ( 1 ) by a constant factor 10. (In this calculation the sampling of the excited states in the primary fragments was carded through in a dif-
Volume 318, number 3
PHYSICS LETTERS B
9 December 1993 131Xe ,before evaporation
13tXe*, Multi f r a g m e n t a t i o n 1.0
. . . .
,
. . . .
,
0.9 .~
0.8
. . . .
,
10.0
. . . .
.
.
.
.
- - Multifragm., formula(B) . . . . Multifragm.,Bethe • .... .., *
...
'
.
.
.
.
'
4
- - formula(8) - - - Bet.he
"""
..... Bethe*lO
0.7 ~
0.6
~
0.5
......""
I 5.0
......
.."
//
~
0.4 ...." f / -~
0.3
o.2 0.1 0.0
500
1000
1500
0.0
2000
0
500
Fig. 2. Relative yield of multifragmentation events of 13tXe. (at least three fragments with A >I 10) as function of the excitation energy for three different level-densities of the excited primary fragments. As can be clearly seen the Bethe formula ( 1) suppresses the multifragmentation. ferent way than described in refs. [2,3]: They were sampled proportional to the density o f excited states and the evaporation of secondary neutrons, protons, and a-s was calculated with the evaporation code Julian [4]. This will be published in a separate publication [5 ]. The resulting fragment distributions are close to the ones o f the old B e r l i n - M M M C model as described in refs. [2,3,6] where M A C was build in explicitly from the beginning.) Evidently the yield o f multifragmentation is suppressed if the Bethe-formula is taken for the internal level-density of primaries at freeze-out instead of the - in this respect - more correct formula (8) which takes account of M A C . The enhancement of the very asymmetric splitting can be seen in fig. (3) in an enhancement of the production of He-isotopes. The corresponding information is given by fig. (4), where the mass of the three largest fragments is given for the decay of 131Xe* as function of the excitation energy. Again we see that the Bethe formula ( 1 ) gives considerably larger heaviest fragments. If the suppression o f the split into more or less symmetric fragments may be "corrected" by simply multiplying the Bethe level-density with a factor 10, the three heaviest masses and also the multifragmentation yield (fig. 2) reproduce roughly the result which uses the formula (8) and obeys M A C . Only at excitation energies above ~ 600 MeV the simple correction ( B e t h e . 10)
1000
1500
Energy (MeV)
Energy (MeV)
Fig. 3. Number of He-isotopes produced in the fragmentation of t3]Xe as function of the excitation energy. The full line is the calculation using the level-density according to formula (8), the dashed line gives the multiplicity of He using the Bethe formula (1), and the dotted curve shows the result of a simulation using the "corrected" Bethe formula B e t h e • 10. 131Xe, three largest fragments Formula(8)-Bethe, a=A/16 .... . .... . .... . ....
130 120
--
110
•
formula
(8)
---- Bethe
""
ioo 90 80
"-"N,
70 r~.
80
"\" k
50
.~ 40
+N,,,
30
",.
20 10 0
--i--0
500
1000
1500
2000
Energy (MeV) Fig. 4. The three largest masses of fragments produced in statistical fragmentation of 131Xe as simulated by microcanonical Metropolis Monte Carlo ( M M M C ) vs. excitation energy. gives too much fragmentation (too small heaviest fragments). Here the heaviest fragments have a mass ~ 50 and the suppression factor o f R ( A = 50, Al, E* ) is only ~, 1/5. Thus multiplying the Bethe formula by 10 enhances the fragmentation too much at higher excitations and the fragments come out too small. Correspondingly, the production of Heisotopes is strongly enhanced further (see fig. (3)). 407
Volume 318, number 3
PHYSICS LETTERS B
9 December 1993
Folding densities 60
i
E * = / O 0 Meg, a = A / 1 6 i i i i
i
i
,-.5O
40
20 /0
~ : 5 0
A=IO0 - - Besse l, {formula(8)] • asympt.Bessel • Bethe*e8
A1.='~2
1'0 ' 2~0 ' 3'0 ' 4~0 ' 5 0 ' 6~0 ' 7~0 ' 8~0 ' 9~0 ' / 0 0
e{MeV} Fig. 5. The integrand of the folding integral eq. (6) leading to R (A, Al, E*) as a function of e for the split of a nucleus with mass A -. A1 = 2, 10,25,50. One may think the failure of the Bethe formula to satisfy M A C as being due to the asymptotic approximation (saddle-point approximation in the Laplace back-transform (3) ). This is certainly the real origin of the problem but is not so straight forward: To visualize the problem, we assume for simplicity that the canonical partition sum Z (A, T) is given by the following Fermi-gas formula which is valid provided the conditions listed above are satisfied: Z ( A , T ) = c ear .
(7)
The single Laplace-back transform of (7) can be done exactly and yields, as discussed by Krappe [7 ], Z(A,E)
= J(E +0) +
I~(2v/~),
(9)
Fig. (5) shows the integrand of the folding eq. (6) as a function of e for the total mass A = 100 split in four different ways: .41 = 2, 10, 25, 50. Apart from the divergence of the asymptotic form at small energies e which can easily be removed (regularized) the agreement with the exact result is excellent. I.e. the asymptotic (saddle-point) approximation for the single Laplace back-transform is not the origin of the failure of M A C . Evidently the discrepancy arises when several Laplace back-transforms are done subsequently. It is the grandcanonical two-particle origin of the Bethe formula (1) that leads to problems.
(8)
where Il (x) is the modified Bessel-function of first order. This result is, however, only correct for low excitation energies (see condition (2) above). In general for excitations where multifragmentations are important the assumption o f Z (A, T) oc eaT is not valid, cf. ref. [2,6] and the free energy is not quadratic in T, see e.g. the calculation of the caloric relation T (E) of ref. [2 ]. We can compare this with the above saddle-point approximation of the corresponding single Laplaceback transform which is of course the well known asymptotic form of the modified Bessel function: 408
al/4 e2v'~ . pcan,asympt " 27tl/2E3/4
3. Conclusion The constraints M A C on the dependence on E* and A of the correct level-density of a large, but finite nucleus with realistic size are quite severe. The leading, volume term of the leptodermous expansion of the level-density is different from the Bethe formula, which is correct only in the asymptotic limit of large mass and energy. This difference has noticeable - and measurable - consequences on the fragment size distribution of statistical mulfifragmentation.
Volume 318, number 3
PHYSICS LETTERS B
Acknowledgement We thank A. Ecker for his help in clarifying the combinatoric problem of the exact level-density of the equidistant model.
References [ 1 ] Aa. Bohr and B.R. Mottelson, Nuclear Structure (W.A. Benjamin, New York, 1969). [2] D.H.E. Gross, Rep. Prog. Phys. 53 (1990) 605. [3] D.H.E. Gross and K. Sneppen, submitted to Nucl. Phys. (1993).
9 December 1993
[4] H. Rossner, D. Hilscher, D.J. Hinde, B. Gebauer, M. Lehmann, M. Wilert and E. Mordhorst, Phys. Rev. C 40 (1989) 2629. [5] R. Heck, D.H.E. Gross and A.R. DeAngelis, in preparation (1993). [6] D.H.E. Gross, in Proc. Second European Biennial Workshop on Nuclear Physics (Megeve, France, March 29-April 2, 1993), ed. D. Guinet (World Scientific, Singapore, 1993). [7 ] H.J. Krappe, in: Dynamical Aspects of Nuclear Fission, eds. J. K_ristak and B.I. Pustylink (Institute of Physics, SASc, Bratislava 1991) p. 51.
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