294
R. Vogt et al.
2 Introduction to FREYA
As input, FREYA requires the mass distribution of the primary fission fragments,
Y (A), and the mean total kinetic energy for a given mass split, TKE(A), for the
particular excitation considered. (FREYA can simulate both neutron-induced fission
and spontaneous fission.)
The initial nucleus A 0 splits into light and heavy fragments, A L and A H ,
respectively. The Q-value for a particular split is Q = M 0 c 2 − M L c 2 − M H c 2 .
The total fragment kinetic energy, TKE, is sampled from TKE(A H ) and the total
excitation energy available for rotational and statistical excitation at scission is
E ∗
sc = Q − TKE. The corresponding “scission temperature” T sc is obtained from
E ∗
sc = a(A 0 )T 2
sc where the scale of the level density parameter a(A) = A/e 0 is
governed by e 0 ≈ 10 MeV. This is the first adjustable parameter in FREYA.
In addition to any overall rigid rotation, which imparts mean angular momenta to
the two fragments, they also acquire fluctuations around the mean values from the
wriggling and bending modes. The magnitude of these spin fluctuations is governed
by the “spin temperature” T S = c S T sc which can be adjusted through the second
FREYA parameter c S . The spin fluctuations vanish for c S = 0.
After subtracting the rotational energy of the two fragments, E rot , a total of
E stat = E ∗
sc − E rot is left for statistical excitation which is distributed between
the two fragments. A preliminary partition, E stat = ´
E ∗
L + ´
E ∗
H , is made according
to the heat capacities of the fragments, which in turn is assumed to be proportional
to the level density parameters, i.e. ´
E ∗
L : ´
E ∗
H = a L : a H . If the shell corrections
are negligible, or the available energy is large, a i ≈ A i /e 0 . Because the observed
neutron multiplicities for known nuclei at low energies suggest that the light
fragments tend to be disproportionately excited, the light fragment is given a larger
excitation energy by the third parameter x, E
∗
L = x ´
E ∗
L , E
∗
H = E stat − E
∗
L , where
x > 1.
After the mean fragment excitation energies have been assigned, FREYA considers thermal fluctuations in the statistical excitation. The mean fragment excitation is
related to its temperature T i by E
∗
i = a i T 2
i with associated variance σ 2
E i
= 2E
∗
i T i .
An energy fluctuation δE ∗
i is sampled from a truncated normal distribution of
variance 2cE
∗
i T i and the fragment excitations are adjusted accordingly, E ∗
i =
E
∗
i + δE ∗
i , i = L, H . Energy is conserved by making a compensating opposing
fluctuation in TKE, TKE = TKE−δE ∗
L −δE ∗
H . The factor c multiplying the variance
is the fourth FREYA parameter. It compensates for the truncation of the normal
distribution due to energy conservation. Finally, TKE may be adjusted by the fifth
and final FREYA parameter, dTKE, to reproduce the average neutron multiplicity,
ν.
The neutrons are evaporated isotropically in the frame of the emitting fragment,
apart from a slight flattening due to the nuclear rotation. Their energy is sampled
from a black-body energy spectrum, dN n /dE n ∼ E n exp(−E n /T max ), where T max
is the maximum possible temperature in the daughter nucleus. FREYA generally
assumes that neutron evaporation continues until the nuclear excitation energy is
R. Vogt et al.
2 Introduction to FREYA
As input, FREYA requires the mass distribution of the primary fission fragments,
Y (A), and the mean total kinetic energy for a given mass split, TKE(A), for the
particular excitation considered. (FREYA can simulate both neutron-induced fission
and spontaneous fission.)
The initial nucleus A 0 splits into light and heavy fragments, A L and A H ,
respectively. The Q-value for a particular split is Q = M 0 c 2 − M L c 2 − M H c 2 .
The total fragment kinetic energy, TKE, is sampled from TKE(A H ) and the total
excitation energy available for rotational and statistical excitation at scission is
E ∗
sc = Q − TKE. The corresponding “scission temperature” T sc is obtained from
E ∗
sc = a(A 0 )T 2
sc where the scale of the level density parameter a(A) = A/e 0 is
governed by e 0 ≈ 10 MeV. This is the first adjustable parameter in FREYA.
In addition to any overall rigid rotation, which imparts mean angular momenta to
the two fragments, they also acquire fluctuations around the mean values from the
wriggling and bending modes. The magnitude of these spin fluctuations is governed
by the “spin temperature” T S = c S T sc which can be adjusted through the second
FREYA parameter c S . The spin fluctuations vanish for c S = 0.
After subtracting the rotational energy of the two fragments, E rot , a total of
E stat = E ∗
sc − E rot is left for statistical excitation which is distributed between
the two fragments. A preliminary partition, E stat = ´
E ∗
L + ´
E ∗
H , is made according
to the heat capacities of the fragments, which in turn is assumed to be proportional
to the level density parameters, i.e. ´
E ∗
L : ´
E ∗
H = a L : a H . If the shell corrections
are negligible, or the available energy is large, a i ≈ A i /e 0 . Because the observed
neutron multiplicities for known nuclei at low energies suggest that the light
fragments tend to be disproportionately excited, the light fragment is given a larger
excitation energy by the third parameter x, E
∗
L = x ´
E ∗
L , E
∗
H = E stat − E
∗
L , where
x > 1.
After the mean fragment excitation energies have been assigned, FREYA considers thermal fluctuations in the statistical excitation. The mean fragment excitation is
related to its temperature T i by E
∗
i = a i T 2
i with associated variance σ 2
E i
= 2E
∗
i T i .
An energy fluctuation δE ∗
i is sampled from a truncated normal distribution of
variance 2cE
∗
i T i and the fragment excitations are adjusted accordingly, E ∗
i =
E
∗
i + δE ∗
i , i = L, H . Energy is conserved by making a compensating opposing
fluctuation in TKE, TKE = TKE−δE ∗
L −δE ∗
H . The factor c multiplying the variance
is the fourth FREYA parameter. It compensates for the truncation of the normal
distribution due to energy conservation. Finally, TKE may be adjusted by the fifth
and final FREYA parameter, dTKE, to reproduce the average neutron multiplicity,
ν.
The neutrons are evaporated isotropically in the frame of the emitting fragment,
apart from a slight flattening due to the nuclear rotation. Their energy is sampled
from a black-body energy spectrum, dN n /dE n ∼ E n exp(−E n /T max ), where T max
is the maximum possible temperature in the daughter nucleus. FREYA generally
assumes that neutron evaporation continues until the nuclear excitation energy is