Shlomo (pdf)

where B

s

is the binding energy of cluster s and ε * i

() s is its excitation energy.

Using Eqs. (1), (7) and (10), we obtain that

( μ

( )) / 1

nNs+ μpZs−Bs−EC s Ms

T

transl. *

M s

Z = ∑∏e ( Qs

Qs)

=

M !

where

{ Ms

} { s}

s

{} s

μs

s/ transl. *

( M T

e Qs

Qs)

= ∏ exp ,

(11)

*

s

i

( )

* ( s)/

T

∑ 2

i( ) 1 − .

(12)

Q = I s + e ε

For the purpose of simplification we have introduced in Eq. (11) the quantity

i

μ = μ + μ − − (13)

s nNs pZ (),

s

Bs EC s

It is important to note that apart from the Coulomb energy EC

() s the quantity μ

s

defined in Eq. (13), is similar to the

cluster chemical potential introduced by ACCR [1] under the condition of chemical equilibrium. We emphasize that, in

contrast to Ref. [1], only the thermal equilibrium condition was imposed in the derivation of Eq. (11). Using Eqs. (3),

(4), (8) and (11), the average multiplicity M

s

of clusters s is given by

1 1

V ′

M M e Q Q e A Q .

∑

∏

( )

μ / transl. * s

sMs T

M

μs/

T 3/2 *

s

=

s s s

=

3 s s

Z { M }

!

s s M

s

λT

(14)

Let us introduce the average density n

s

of clusters s :

M

s

ns ≡ n( Ns, Zs) = .

(15)

V

Using Eqs. (12) - (14) one finds for the nucleon densities n

n

and n

p

the expressions,

2χ

1 2χ

1

p / T

nn

= e , n = e μ

,

1+ κλ

1+

μ n / T

3 p

3

T

κλT

(16)

where χ = V′

/ V0

is the hindrance factor. Note that the nucleon spin-degeneracy factor 2 was taken into account. From

Eqs. (13) - (16) one finds for the relative yield of fragments s the expression

n

n

s

Ns

Zs

n

np

1⎛1+

κ ⎞

= ⎜ ⎟

2⎝

2χ

⎠

As

−1

A

( λ )

3/2 3

As

−1

*

s T s

The ACCR method

Q e

− ( Bs+

EC( s))/

T

.

(17)

In the ACCR approach, the chemical equilibrium condition has the form,

μ( AZT , , ) = Zμ ( T) + ( A− Z) μ ( T) + B( AZ , ),

(18)

p

Here, μ ( A, ZT , ), μ p( T ) , and μ ( T ) n

are the chemical potentials of the fragment ( A, Z ), the free proton and neutron at

the temperature T , respectively, and BAZ ( , ) > 0 is the ground state binding energy [17] of the fragment ( A, Z ).

Employing Boltzmann statistics, the temperature can be deduced from the double ratio

Y( A′ , Z′

)/ Y( A, Z )

R = = F A, Z , A, Z , A , Z , A , Z ⋅exp ΔB/ T ,

n

( ′ ′ ′ ′ ) ( )

1 1 1 1

2 1 1 1 1 2 2 2 2

Y( A′ 2, Z′

2)/ Y( A2, Z2)

⎛ A′

⋅ A ⎞

F( A′ Z′ A Z A′ Z′ A Z ) = ⎜ ⎟

⎝ ⎠

3/2

( 2 I( A′ 1, Z′ 1) + 1)( 2 I( A2, Z2) + 1)

( 2 ( , ) 1)( 2 ( , ) 1)

1 2

1, 1, 1, 1, 2, 2, 2, 2

,

A1⋅ A′ 2

I A1 Z1 + I A′ 2

Z′

2

+

(19)

where I( AZ , ) and Y( A, Z ) are the total angular momentum of the ground state and ground state yield of the fragment

( A, Z ), respectively. The quantity

Δ B is given in terms of the binding energies of the fragments:

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