Mononuclear Caloric Curve

LGS - WU

ACS (3/13/05)IU – V.V. (10/29/05)

Mononuclear Caloric Curve: DATA

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8

(MeV/amu)

T (M

eV)

100-140

140-180

180-240

Mononuclear Caloric Curve: DATA + Fermi Gas

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8

(MeV/amu)

T (

MeV

) 100-140

140-180

180-240

FG-8

Series2

1) Relation of Thermal energy (U) to Temperature (T) 2aTU .2aTCv aUT / and (as for e- in metals) =>so

Phase transition?

Mononuclear Caloric Curve: IdeaThis model1) is designed to create a better description of the (mononuclear)

density of states appropriate for statistical (kinetic) decay calculations.

2) takes into account, in a plausible fashion, both expansion and many-body correlations.

1) provides a recalibration of the mononuclear (thermodynamics) expectations.

Principle results: 1) One should expect a plateau in the mononuclear CC.

2) The idea of the CN is “expanded” (pun intended) to higher E*

3) But in terms of the real world we must confront VV and then we find …..

Entropy… of a Fermi drop

.)(2)(222'' expexp

*

0

aAEEaaUaTdTT

CS

Tv

Entropy is a function of the “thermal” energy. Must subtract E required for expansion (as one does for collective rotation.)

2aTU

dr

mk

ra

f ]2/)[(

)(

4 *2,

2

4) The local-density approximation lead to a simple expression for “a”. a to the density of SP states (g)

2) With no PT’s

.2aTCv 1) Relation of Thermal energy U to Temperature

aUT /or and

3) As always, the statistical T is vU

ST )(1

With models for “a” and exp => calc S(U) => T(U)

Mononuclear Caloric curve: Overview

Nuclear expansion by maximizing

entropy(Toke et al.)

Expansion + Effective mass logic ~

S. Shlomo, J. Natowitz, et al.

Numerical scheme for little “a” Calculation with m* logic ….“LDA”Barranco and TreinerPrakash, Wambach and Ma

Mononuclear (pseudo) plateau Sobotka et al.

Mononuclear “Caloric Curve”: Intellectual input

1. The nuclear expansion is determined by maximization of the entropy. 1st pass: Self-similar expansion ( c ) - 2nd pass: add surface diffuseness (b,c)

2. Entropy from simple FG expression. (No clustering)

3. The level density constant is determined by a local-density approximation.

4. Many-body correlations enter through effective mass factors, accounts for .. a) the FR (standard effective mass for NM) [mk < 1 , mk => 1 as => 0 ] b) the surface [mw > 1 (sur), mw => 1 as ’ =>

0 ]

5. The thermal energy is the difference between the total excitation energy and the energy required for expansion.

6. The expansion energy is determined either by:a) the simple form suggested by Friedman (EES) b) that suggested by Myers and Swiatecki (TF)c) DFT (light – a la’ van der Waals)

7. Statistical Temperature from differentiation of entropy w.r.t. E*: 1/T = (dS/dE*)v

Native radial profile

r2

2

2

)*(

2/1

4

)2(b

Rcr

oo

eb

cdF

))(7071.0

(1{2

),(b

Rrerfrbf oo

),(),,( 3 rcbfcrcbF

Self-similar expansion

Gaussian Derivative

Family of drop shape profiles

0 2 4 6 8 10 12 14

0.05

0

0.05

0.1

0.150.16

0.06

f A 1 x( )

F A 1 0.9 x( )

F A 1 0.8 x( )

dF A 1 1 x( )

F A 2 1 x( )

140 x

(r) [#/fm3]

r (fm)

1/c = linear expansion b = surface diffusness

Need for m*Designer fix to get single-particle E’s = >• ~ Corrects for momentum

dependence of interaction.• Can be viewed as a non-locality in

SPACE (a r-p “thing”) • It is caused by the “finite range” of the

interaction.• If you want a simple local “potential”

picture you must correct for this simplification.

• The math

][2

)(][

2

)( 22

EVm

kE

dE

dVV

m

kE oo

oVm

kE

2

)()1(

2

)1()1(2

)( 2

oV

m

kE

'*

2

2

)(oVm

kE

7.0~)1(~)1()1( 1*

dE

dV

dE

dV

m

m

Many-body correlations via Effective mass“k-mass” – (mk) and “omega-mass” – [mω]

momentum dependent interaction “single-particle” to surface coupling

Accounts for:the true p dependence of the potential

or (through FT)the r-nonlocality of the true interaction

Needed for:a) “single-particle” energies - SE solutions of LOCAL potential problem.

mk < 1

In principle depends on density [and asymmetry [(n-z)/A].

HOWEVERAs an “mk”<1is needed to get SP E’s…This factor REDUCES the density of both SP and MP states.

Accounts for:the true E dependence of the potential

or (through a FT) the t-nonlocality of the true potential.

Needed for:a) “multiparticle” Density of States

or its log the Entropy => mω > 1

b) reduced spectroscopic strength near the Fermi surface.

Can be viewed as:a coupling of collective (surface) excitations with “single-particle” excitations

+The result of SRC

Effective mass - summary

Expansion:Returns mk to 1 (as ρ => 0)Returns mw to 1 (as ρ’ => 0)

Heating: Returns mw to 1.(as collective

surface modes die.)

]'

1)[1(])[(*

oowk mm

m

m

0 2 4 6 8 10 12 140.6

0.8

1

1.2

1.4

mkw 1 1 r( )

mkw 1 0.9 r( )

mkw 1 0.8 r( )

mkw 2 1 r( )

r

Expanding

0 2 4 6 8 10 12 140.6

0.8

1

1.2

1.4

m 1 1 r 0.5( )

m 1 1 r 2( )

m 1 1 r 4( )

r

Heating

r

r

m*

Expansion Energyeb To a 2

b 3 c 5

0 0.2 0.4 0.6 0.8 1 1.220

15

10

5

0

5

1010

20

eb x 0( )

eb x .2( )

eb x .4( )

eb x .6( )

1.40 x0 0.2 0.4 0.6 0.8 1

0

5

10

12

1

BEexpn x 0.2( )

BEexp x 0.2( )

coule x( )

Eexpn x 0.2( )

Eexp x .2( )

F x( )

10 x

Myers and Swiatecki EoS similar to Friedman (EES) formCoulomb modified => expansion E = b(1 - )2

for small density reduction

Eexpansion

BE

Coul

NUCTot

EES

BENM(c,

Expansion Compression => Expansion

symm

etric

asymmetric

Mononuclear “Caloric Curve”

Results for:expansion ( c )

wo m* b= 8,

and with m* for b= 8 and b= 6

0.8 0.9 1.0

10

30

1234

5

67

SE

C

M

0 1 2 3 4 5 6 7 8

3

4

5

6

(MeV)

T(M

eV

)c)

8

1

1.5

0

0.05

0.1

0.15

0.2

sM

s M

E0.5

2

E (MeV) =

b)

8

8

0.4

0.6

0.8

1

a)

8

The “go of it” - Level Density parameter (a)

a) mk provides +ve feedback on expansion! b) mw drops and initially pulls down “a” then c) expansion and mk take over and increase “a”

0 1 2 3 4 5 6 7 820

25

30

35

4040

20

a o C o x( ) a k C 1 x( ) a kw C o x( ) a C 1 x( ) x

80 xE*/A

a (#/MeV)

(MeV/amu)

Remember a SP “level density” at Fermi surface = g

First-pass conclusions1. This is a repackaging of what Natowitz and Shlomo have been

saying.

2. Its advantage over the previous work is that the approach is straightforward.The mathematics is well posed and the solution predicts:an ~ plateau in the caloric curve for a mononucleus resulting from a) reduction in density and b) the destruction of collective effects.

3. At some point in the plateau, the multi-fragmented (MF) density of states exceeds the mononuclear value. Where this happens depends on the volume used for MF. HOWEVER as VMF > VMN, the MN can (still) be considered meta stable. HOWEVER there MUST be a transient delay (PS exploration time.)

4. The logic is simple AND restricted to the time window for which an equilibrated mononucleus is reasonable. However as the Temperature saturates so do the decay rates.

GREATER ContextDFT landmarks:1873 – J.D. van der Waals, Thesis DFT - light1964 – P. Hohenberg and W. Kohn [PR 136B, 864 (1964)] Real DFT1965 – D. Mermin’s [PR 137, A1441 (1965)] Finite T DFT

Generally appreciated that the gradient expansion does NOT converge for induced dipole (u~r-6).

See Rowlinson and Widom, Molecular theory of Capillarity, but (Oρ)2 works damn well!On the other hand, expansion justified for pair potential = Yukawa.

Nuclear (EDF)1973 – R. Lombard [Ann. Phys 77,380 (1973)]. (Oρ)2 ,(Oδ)2

1985 – J. Bartel, M. Brack and M. Durand NP A445, 263 (1985)(Oρ)2,4 and Wigner-Kirkwood expansions + m* (formally!)

1990 – P. Gleissel, M. Brack, J. Meyer and P. Quentin, Ann Phys. 197, 205 (1990)]Approach: Paramatierized shape + variational => GR and validity of Lepdoderous (DM) expan.

FHelm [ρ], KE [ρ], Entropy [ρ], Jso [ρ] However m*(r,E*) is not accounted for. (In principle YES in practice

NO.

2005 – include mkmw decomposition (in a simplified model)

“EDFT” - light

0

2*),,(*),(4

),( drrrcbFeA

cbE bmd

1. Energy of “matter drop”

3. Gradient2 Correction (gc parameter)

4. Sum with 3D Coulomb integral

0

222

*)),,((8

**4),( drrrcbF

Am

gccbE

ngt

coulgtmdDFT EEEcbE ),(

2. Coulomb Energy (exchange also)

0

23 )4(1)

3

4(),( eecoul drr

rrcbE

Agrees with LDM to ~ 4 ppt from A = 10-250

Binding EnergyOn beta stability for( b = c = 1)

BE(MeV/amu)

50 100 150 2007.5

8

8.5

9

1

0.5

0

0.58.763

7.5

BE x( )

DM x b x( )

0.5

1.0

Rd x( )

24010 x

E

D (M

eV/u

) 50 100 150 200

A

% D

iffere

nce

7.5

8.0

8.5

9.0

-1.0

-0.5

0.0

0.5

A (amu)

Moving on: Eexp(b,c)

b: surface => y

c: bulk expansion

=> x

Entropy (per nucleon) for = 2,3,4,5 MeVs(c,b) for m* = 1

2D analysis flatter than 1D analysis, which is flatter than FG

0

2

4

6

8

0 1 2 3 4 5 6 7

(MeV/u)

T(M

eV

)

But can this model reproduce Vic’s fragment energies?

CalculateK = B(b,c)/B(1,1)

Compare Keq to Vic’sK = experimental Barrier reductions

ANS = NO !0 1 2 3 4 5 6 7 8

0.6

0.7

0.8

0.9

1

1.11.1

0.6

K Vc

K Vu

K Vl

K df

7.90 K Ve K Ve K Ve K dfe0 1 2 3 4 5 6 7 8

(MeV/u)

Keq

0.6

0.7

0.8

0.9

1.0

Bar0.8 0.9 1.0 1.1 1.2c

1.0

2.0

3.0

b

Should the coefficient of the gradient term be temperature dependent ? bg2(T,ρ) = ?

Second view: WEAK!

V

g rdbFA

cbF~

222

1___

...})({4

),(

V

g rdCrT

b~

22 }{

6

]1)[(...)]23()

13()

12([

3)

12( )( ru

VP

ergrfrfrfdrrfC

Formal view:

WEAKTbc

T

TT

g

)1(~)1(~~)( 03.0

)(2 4

2

29.1)1(~CTT 25.1)1(~

CTT

T34.0)1(~

CTT

The cost of a gradient is:

In terms of the direct correlation function:

Surface tension iso T compressibility Miscibility gap.

Second-pass Conclusions

1. The S analysis implies a “normal-mode” with about 2/3th surface (b) and 1/3rd bulk (c) character.

2. In this model, “Multifragmentation” is then a fluctuation in this mode, creating a “surface” region unstable to cluster formation. SUFACE not VOLUME

3. A plateau in a Caloric Curve ≠ Phase TransitionPrecursor to PT – Yes but …..

In fact, why bother talking about PT at all? Does PT logic really help you understand “the go of it” in the meso world.

I think not.4. BUT the fragment energies indicate the EQ model is not

sufficient as Kexp < K(eq). “Dynamical” expansion is indicated. EES is such a model (although it is completely misunderstood and advertised.)

Mononuclear CC

Caloric Curve

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

epn (MeV)

T (

MeV

)

100-140

140-180

180-240

197

NE-FG-8

NE-FG-10

2D-bc

Poly. (197)

Poly. (NE-FG-8)

Poly. (NE-FG-10)

Poly. (2D-bc)