DRAFT for IAEA CRP (Nucl.Data Sheets)– 11 March 2014
Description of the Point-by-Point model of prompt emission
Anabella Tudora
The primary result of the Point-by-Point (PbP) model consists in the so-called multiparametric matrices of different quantities as a function of fragment mass and charge numbers A, Z
and as a function of total kinetic energy TKE, generally labeled q(A,Z,TKE). These matrices are
referring to both the fission fragments and the prompt neutron and γ-ray emission, e.g. the total
excitation energy at full acceleration TXE(A,Z,TKE), the fragment level density parameter
a(A,Z,TKE), the fragment average neutron separation energy <Sn>(A,Z,TKE), the prompt neutron
multiplicity ν(A,Z,TKE) and spectrum N(A,Z,TKE, E), the prompt gamma-ray energy Eγ(A,Z,TKE).
These matrices do not depend on the fragment distributions Y(A,Z,TKE). The calculation of the
matrices q(A,Z,TKE) needs as input only data of recommended nuclear data libraries (e.g. mass
excesses, deformation parameters, shell corrections, phenomenological optical parameterizations,
usually taken from RIPL3)
1. The fragmentation range of the PbP treatment
In the PbP treatment the choice of fission fragments plays a crucial role. As it was already
mentioned in previous papers (see for instance [1 – 4] and references therein) the range of fission
fragments is constructed as following. For each mass pair {AL, AH}, covering a large A-range from
symmetric fission up to a very asymmetric split with a step of one mass unit, up to four charge
numbers Z are taken as the nearest integer values above and below the most probable charge Zp.
This is taken as unchanged charge distribution (UCD) corrected with the charge deviation
(polarization) ΔZ. As an observation, because the charge distribution for any fixed A is narrow [5],
usually in the PbP treatment the consideration of 2 or 3 charge numbers Z for each A is enough [6].
Prompt emission calculations are done for each fragment (with Z and A obtained as mentioned
above) at TKE values covering a convenient range (e.g. from a minimum TKE taken usually of
about 110-140 MeV up to a maximum TKE, usually of about 200-225 MeV) with a step of 5 MeV.
The fragments taken into account in the PbP treatment (meaning the charge numbers Z at each A)
depends on the most probable charge considered as:
Z p ( A)  ZUCD ( A)  Z ( A)
(1)
Charge deviations as a function of A fitted by Wahl [7] are used for fissioning systems with
experimental ΔZ data (e.g. 233,235U(nth,f), 239Pu(nth,f) and 252Cf(SF)). For other fissioning systems
without any ΔZ information, the consideration of a constant ΔZ = ±0.5 for all A (with + sign for the
light fragments and minus sign for the heavy ones)) is a good approximation [6].
2. Partition of total excitation energy between complementary fully accelerated fragments
In the calculation of different prompt emission quantities q(A,Z,TKE) the excitation energies
of complementary fully-accelerated fragments EL*,H are needed. In the PbP treatment the partition of
TXE  EL*  EH* is obtained from modeling at scission, as it was described in detail in Ref.[8] or by
using parameterizations based on the systematic behaviour of the experimental ratio νH/(νL+ νH) as a
function of AH as described in Ref.[9].
1
For each pair of fragments of the fragmentation range, TXE is taken as
TXE  Er  En  Bn  TKE in which the energy release Er (Q-value) is calculated using mass
excesses from nuclear data libraries (usually of Audi and Wapstra taken from RIPL3 [10]), En is the
incident neutron energy and Bn the neutron binding energy of the fissioning nucleus. Obviously in
the case of spontaneous fission both En and Bn are taken zero.
The modeling at scission starts from the energy conservation at scission for each pair of
nascent fragments:
L
H
(2)
Er  En  Bn  E pre  ECoul  Esc*  Edef
 Edef
where Epre and ECoul are the pre-scission kinetic energy and the Coulomb repulsion energy between
the two nascent fragments, respectively. E sc* is the available excitation energy at scission. At full
acceleration E pre  ECoul  TKE and
L
H
TXE  Esc*  Edef
 Edef
(3)
L,H
in which Edef
are the so called extra-deformation energies (or additional deformation energies)
according to Refs. [8, 11]. These extra-deformation energies represent the differences between the
fragment deformation energies at scission and at full acceleration and are determined (in the first
calculation step) from the liquid drop model with shell corrections taken into account [8]. In the
second calculation step E sc* (obtained by subtracting the extra-deformation energies from TXE) is
shared between complementary nascent fragments, assuming the statistical equilibrium at scission
and a Fermi-gas description of fragment level densities, according to the ratio
E scL E scH  a scL a scH
(4)
in which a scL , H must be understood as effective level density parameters accounting for collective
and intrinsic/single-particle excitations(for details see Ref.[8]). The values of excitation energy at
scission E scL . H and of level density parameters at scission a scL , H are obtained simultaneously by an
iterative procedure according to Eqs.(3, 4) and the level density parameters given by the super-fluid
model of Ignatiuk [12]:
 ~  W ( Z , A)

*
a ( A) 1  U * (1  exp(  ( A)U *)  U  U cr
a ( Z , A, E*)  


a
U *  U cr
 cr
U *  E *  Econd
(5)
in which the notation E* means the excitation energy at scission E scL . H . In Eq.(5) the condensation
energy is Econd  3acr 2 2 2  n , where acr is the critical level density parameter,   12 A is
the pairing correlation function and n = 0, 1 and 2 for even-even, odd-A and odd-odd nuclei,
respectively. The critical temperature of the phase transition from super-fluid (super-conductive) to
normal states is given by tcr  0.567 and the critical energy is Ecr  acr tcr2 . W is the shell
correction, a~ ( A) is the asymptotic value of the level density parameter (obtained when all shell
effects are damped) and γ(A) is the parameter of the function defining the damping of shell effects.
Different shell corrections (such as those of Möller and Nix [13] or those calculated with the Myers
and Swiatecki mass formula taken from RIPL3 [13]) and different parameterizations of the dumping
and the asymptotic level density parameters can be used. In the majority of PbP calculations the
parameterizations proposed by Ignatiuk [12] a~  0.073 A  0.115 A2 / 3 and   0.4 A1/ 3 are used
2
together with the shell corrections of Möller and Nix. Other a~ and γ parameterizations (e.g. those
given in Ref. [14]) can be used, too.
Finally the excitation energies of complementary fully accelerated fragments are obtained as
a sum of extra-deformation and excitation energies at scission:
L, H
(6)
EL*,H  Edef
 EscL,H
The shape of the ratio EH* TXE as a function of AH obtained from this modeling at scission
is similar to the ratio  H ( L  H ) obtained from experimental ν(A) data. This systematic
behaviour for neutron induced fission (at low and moderate En) consists in the following aspects:
the ratio is less than 0.5 for fragment pairs with AH < 140 with a minimum at AH around 130 (due to
the magic or double magic heavy fragments with N=82 and/or Z=50). It is of about 0.5 for AH
around 140 and it exhibits an almost linear increase for pairs with AH above 140 (details are given in
Ref. [9]).
3. Prompt emission of individual fragments
For a fully accelerated fragment a Weisskopf-Ewing prompt neutron evaporation spectrum
in the center-of mass system (CMS) at a given residual temperature T (  ( )  c ( )  exp(   T ) ) is
taken. The compound nucleus cross-sections of the inverse process of neutron evaporation σc(ε) of
the nuclei forming the fragmentation range are obtained from optical model calculations (using the
code SCAT2 [15]) with phenomenological potential parameterizations appropriate for nuclei
appearing as fission fragments (usually of Becchetti-Greenless, Koning-Delaroche [16]). To account
the excitation energy distribution, a residual temperature distribution P(T) is used, leading to the
following prompt neutron spectrum in CMS for an individual fragment:
Tm
( )   c ( )   k (T ) P(T ) exp(  / T ) dT
(7.1)
0
with the normalization constant
1


(7.2)
k (T )     c ( )  exp(  / T ) d 
0

For each fragment the maximum temperature value Tm in eq.(7.1) is obtained from
EL*,H  aL,H Tm2L,H in which EL*,H are the excitation energies at full acceleration given by eq.(6) and
aL,H are their level density parameters given by the relation (5) in which, this time, E* is the
excitation energy at full acceleration. For P(T) different functions can be used, e.g. the triangular
form proposed by Madland and Nix [17-19], the form proposed in Ref.[20] (driven by the parameter
s ≥ 1) or numerical functions resulted from a rigorous treatment of the sequential emission.
The most important emission of prompt neutrons takes place at the full acceleration of
fragments. But neutron evaporation during the fragment acceleration can be also possible and these
neutrons can lead to a non-isotropic neutron spectrum in CMS. Another possible source of nonisotropic neutrons can be the emission at the moment of scission (so-called scission neutrons). The
assumption of an anisotropic neutron emission leads -in some cases- to a better agreement of the
calculated prompt neutron spectrum with the experimental data. It was assumed (see Ref.[20] and
references cited therein) that the anisotropy of neutron emission, if present, is symmetrical about 90o
and the prompt neutron spectrum in CMS could be described as:
3
1  b cos 2  cm
(8)
( , cm )  ( )
1 b / 3
where ( ) is given by eq.(7.1) and b is the anisotropy parameter.
The CMS prompt neutron spectrum is transformed into the laboratory frame LS (using the
well-known relation E    E f  2  E f cos cm from which cosθcm is extracted and replaced in
eq.(8)) leading to the following general expression of the prompt neutron spectrum of an individual
fragment A, Z at a given TKE:
( E  Ef ) 2
N ( A, Z , TKE, E ) 

( E  Ef )
( )
2
1  b( E    E f ) / 4 E f
4  E f (1  b / 3)
d
(9)
in which ( ) is given by eqs.(7) and Ef is the average kinetic energy per nucleon obtained from
momentum conservation for a given pair of fragments:
A TKE
E f L,H  H ,L
(10)
AL , H A0
where A0 is the mass number of the fissioning nucleus. The PbP computer code is working with the
general formula (9) allowing different P(T) expressions and to take or not into account the
anisotropy (controlled by the input value of the b parameter). The prompt neutron spectrum in LS
corresponding to a pair of fragments at a given TKE is:
r
1
N pair ( E ) 
N L (E) 
N H (E)
(11)
r 1
r 1
where NL,H(E) are calculated according to eq.(9) and r is the ratio of neutrons emitted by
complementary fragments r  L  H . The prompt neutron multiplicity of a fragment pair
 pair   L   H is obtained from energy conservation:
TXE   pair      Sn    E  pair
(12)
where <ε> is the average prompt neutron energy in CMS (first order momentum of prompt neutron
spectra of complementary fragments given by eqs.(7)), <Eγ> is the average prompt γ-ray energy
and <Sn> is the average neutron separation energy of complementary fission fragments.
The average neutron separation energy of each fragment is calculated by an iterative
procedure accounting for sequential emission. If the excitation energy of a fully accelerated
fragment E*(A,Z,TKE) received from the TXE partition (meaning EL*,H of eq.(6)) exceeds the
neutron separation energy Sn(A,Z) of the respective fragment then a neutron can be evaporated (in
competition with γ-ray emission). If the excitation energy of the residual fragment exceeds it
neutron separation energy Sn(A–1,Z) a second neutron can be evaporated and so on. A number of k
neutrons can be emitted when the excitation energy of the (k-1)-th residual nucleus becomes less
than Sn(A–k,Z). The average neutron separation energy when k neutrons can be emitted is taken as:
1 k 1
1
 Sn  ( A, Z )   Sn( A  i, Z )  Skn( A, Z )
(13)
k i 0
k
in which Sn(A–i,Z) are obtained using the mass excess of nuclear data libraries (usually of Audi and
Wapstra [10]).
In the PbP model the linear correlation between the average prompt γ-ray energy and prompt
neutron multiplicity of complementary fragments is taken into account. The slope of this correlation
4
expresses the competition between the prompt neutron and γ-ray emission at excitation energies
above the neutron separation energy and the intercept expresses the γ-ray decay at excitation
energies below the neutron separation energy. The global parameterization of the slope and intercept
was deduced in 2000 [21-23] by fitting the results of the (n,n’) and (n,γ) competition obtained from
nuclear reaction calculations: direct interaction mechanism treated with spherical optical model and
compound nucleus mechanism by statistical Hauser-Feshbach model with HRTW fluctuation
corrections (code STATIS). This competition was studied for a limited number of nuclei appearing
as fission fragments (given by the “7 point” approximation) in the thermal neutron induced fission
of main actinides and the spontaneous fission of 252Cf and 236-244Pu.
4. Fission fragment distributions and average quantities
To obtain different average quantities as a function of A, as a function of Z, as a function of
TKE and total average quantities (needed in the comparison with the available experimental data for
validation and also for evaluation purposes) the matrices q(A,Z,TKE) are averaged over the fragment
distributions in different manners as following:
q( A) 
 q( A, Z ,TKE)Y ( A, Z ,TKE) Y ( A, Z ,TKE)
Z ,TKE
Z ,TKE
 q  (TKE )   q( A, Z , TKE )Y ( A, Z , TKE )
A, Z
q( Z ) 
(14.3)
A,TKE
 q( A, Z ,TKE)Y ( A, Z ,TKE) Y ( A, Z ,TKE)
A, Z ,TKE
(14.2)
A, Z
 q( A, Z ,TKE)Y ( A, Z ,TKE) Y ( A, Z ,TKE)
A,TKE
 q 
Y ( A, Z ,TKE)
(14.1)
(14.4)
A, Z ,TKE
The PbP results of q(A,Z,TKE) can be averaged over any experimental or theoretical Y(A,Z,TKE)
distribution. The use of experimental Y(A,TKE) distributions is preferred. Usually these are provided
as single distributions Y(A), TKE(A) and σTKE(A). The charge distribution for any fixed mass is taken
as Gaussian [5], the Y(A,Z,TKE) distributions being obtained as:
 (TKE  TKE( A)) 2  1
 ( Z ( A)  Zp( A)) 2 
Y ( A)

 

(15)
Y ( A, Z , TKE) 
exp  
exp
c
2( TKE ( A)) 2  c
 TKE ( A) 2



in which Zp(A) is given by Eq.(1) and c  2( z2 ( A)  1/ 12) or c  2 (rms( A)) 2 when root-meansquare data are available [7].
The PbP treatment can be used also in the frame of the multi-modal fission. In this case the
primary matrices of different quantities q(A,Z,TKE) are averaged over the multi-modal distributions
Ym(A,Z,TKE) (“m” meaning the fission mode) according to eqs.(14). Total average modal quantities
(referring to both, fission fragments and prompt neutron emission, e.g. <Er>m, <TXE>m, <a>m,
prompt neutron multiplicity <ν>m and spectra Nm(E)) were reported in Refs. [20, 24-26].
The total average values of fragment quantities <Er>, <TKE>, <Sn> and <a> obtained by
averaging the corresponding PbP matrices over the Y(A,Z,TKE) distributions according to eq.(14.4)
can be used as input parameters of the Los Alamos model of Madland and Nix [17] (working with
only one fragmentation, the so-called most-probable fragmentation). The systematic of Los Alamos
model parameters reported in Ref.[27] is based on the PbP treatment.
5
5. Examples of PbP results
Examples of different prompt emission quantities q(A,TKE), q(A), <q>(TKE) and total
average ones <q> in comparison with available experimental data are given in Figs.1-5.
Primary matrices of ν(A,TKE) and Eγ(A,TKE) are exemplified in Figs.1a-b for the case of
252
Cf(SF). They are in good agreement with the existing experimental data [28, 29]. Other matrix
results can be found in Refs.[4, 30, 31]
Examples of average prompt emission quantities as a function of A are given in Figs. 2 and
3. The  (A) results of 252Cf(SF) (Fig.2a) and of 235U(n,f) at En=0.5MeV and 5.5 MeV (Fig.2b)
describe well the experimental data taken from EXFOR [32] including the multiplicity increase with
En for heavy fragments only (see the red circles and blue stars in Fig.2b). The E γ(A) results of
233
U(nth,f) and 239Pu(nth,f) (red circles) describe very well the experimental data of Pleasonton [33]
(open squares) as it can be seen in Fig.3a,b. Other PbP results of prompt emission quantities as a
function of A can be found for instance in Refs.[2-4, 8, 9, 34].
Examples of average prompt emission quantities as a function of TKE are given in Figs.4a-b
for <ν>(TKE) and <Eγ>(TKE) of 252Cf(SF) As it can be seen they describe well the existing
experimental data [29, 32]. Other examples of <ν>(TKE) are given in Ref.[1].
Results of prompt neutron multiplicity distribution P(ν) in good agreement with
experimental data are exemplified for 239Pu(nth,f) and 252Cf(SF) in Fig.5. P(ν) of 239Pu(nth,f) (red
circles) gives an excellent description of the experimental data (open squares). Other P(ν) results for
spontaneous and neutron induced fission of actinides were reported in Ref. [30].
Results of total average prompt emission quantities (neutron multiplicity, γ-ray energy and
prompt neutron spectra) were reported in the last 10 years (see for instance recent results for
232
Th(n,f), 234, 238U(n,f) in Refs. [2, 3, 35]). Here we give as example only prompt neutron spectrum
results for 233U(nth,f) and 232Th(n,f) plotted in Figs.6a,b (as ratios to a Maxwellian spectrum) in
comparison with the experimental data (different symbols).
Average prompt emission quantities as a function of Z can be also calculated, according to
eq.(2.3), but experimental data for comparison are missing. An example of a such result is given in
Fig.7 for  (Z) of 235U(nth,f) and 239Pu(nth,f). Proton and neutron even-odd effects are reflected in the
behaviour of  (Z). The staggering in  (Z) is visible for asymmetric fragmentations and it is more
pronounced for heavy fragments.
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6
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7
Cf(SF)
Zackharova
Bowman
AH=135
Prompt -ray energy of fragment pair
Prompt neutron multiplicity of fragment pair
12
252
8
4
0
8
4
0
8
4
0
8
4
0
8
4
0
8
4
0
PbP
AH=146
AH=147
AH=156
AH=159
150
160
170
180
190
200
210
Nifenecker,
4
12
AH=153
AH=150
4
12
AH=147
8
4
12
AH=144
8
Nifenecker,
4
12
AH=141
b
8
Nifenecker,
4
140
220
PbP
PbP
150
160
170
180
190
200
210
b)
TKE (MeV)
a)
TKE(MeV)
Cf(SF)
PbP
8
AH=160
140
252
8
Fig.1: Examples of multi-parametric matrices of 252Cf(SF): a) Average prompt neutron multiplicity and b)
average prompt γ-ray energy as a function of TKE for fragments pairs with AH indicated in figures, PbP
results with full red circles and experimental data with different open symbols.
5.0
4
Zeynalov
EXFOR Bowman 1963 USABRK
EXFOR Zakharova 1999 RUSKUR
PNPI
Budtz-Jorgensen, Knitter 1998
252
Cf(SF)
PbP calculation
3Z/A, Z, rms Wahl
Y(A,TKE) IRMM
235
4.5
Prompt neutron multiplicity
Prompt neutron multiplicity
5
3
2
1
4.0
3.5
PbP 3Z/A, Z, rms Wahl
Y(A,TKE) Straede
0.5 MeV
5.55 MeV
U(n,f)
Mueller, Naqvi GERKFK
0.5 MeV
5.55 MeV
3.0
2.5
2.0
1.5
1.0
0.5
0
0.0
80
90
100
110
120
130
140
150
160
170
A
80
90
100
110
120
130
140
150
160
A
a
b
Fig.2: Examples of ν(A) results for a) Cf(SF) (red circles) and b) U(n,f) at En = 0.5 MeV (red circles)
and En = 5.55 MeV (blue stars) in comparison with experimental data (different black and gray symbols).
252
235
7
8
233
Pleasonton 1973
239
7
Pu(nth,f)
Pleasonton 1973
Prompt -ray energy (MeV)
Prompt -ray energy (MeV)
6
PbP calculation
3Z/A, Z, rms Wahl,
Y(A,TKE) EXFOR
U(nth,f)
5
4
3
2
1
PbP calculation
3Z/A, Z, rms Wahl,
Y(A,TKE) IRMM Wagemans
6
5
4
3
2
1
0
0
80
90
100
110
120
A
130
140
150
80
90
100
110
8
130
140
150
160
A
a
Fig.3: Examples of Eγ(A) results (red circles) for a)
experimental data of Pleasonton (open squares).
120
233
U(nth,f) and b)
239
Pu(nth,f) in comparison with
b
252
Cf(SF)
252
9
Budtz-Jorgensen, Knitter, 1989
Vorobyev, PNPI 2001 RUSLIN
Sing Shengyao 1984 CPRAEP
6
5
Nifenecker et al 1973
4
3
2
PbP calculation
3Z/A, Z, rms Wahl
1
Cf(SF)
8
<E> (MeV)
Average prompt neutron multiplicity
10
7
7
6
PbP calculation
3Z/A, Z,rms Wahl
5
4
0
140
150
160
170
180
190
200
210
140
220
150
160
170
180
190
200
210
220
TKE (MeV)
TKE (MeV)
b
a
Fig.4: Examples of average quantities as a function of TKE for 252Cf(SF): a) <ν>(TKE) and b) <Eγ>(TKE)
PbP results with full red circles in comparison with experimental data (different open symbols).
0.35
0.40
252
239
Pu(nth,f)
0.35
0.30
Holden 1988 USABNL
0.25
(<p> = 2.875 + - 0.009)
0.25
0.20
P()
P()
Vorobyev 2001 RUSLIN
Balagna et al. (1973)
Cf(SF)
0.30
0.20
0.15
0.15
0.10
0.10
PbP calculation
PbP calculation
3Z/A, Z, rms Wahl
Y(A,TKE) IRMM
0.05
(<p> = 2.8787)
0.05
0.00
0.00
0
1
2
3
4
5
6
0
7
1
2
3
4
5
6
7
8
Number of prompt neutrons
Number of prompt neutrons
Fig.5: Examples of P(ν) for a) 239Pu(nth,f) and b)
with experimental data (different open symbols).
a
Cf(SF): PbP results with full red circles in comparison
b
252
233
1.3
1.2
EXFOR
Starostov 83
Starostov 85 (TM)
Boystov 83 (TM)
Lajtay 85
Nefedov 83 (Cf)
Starostov 83 (Cf)
U(nth,f)
1.2
1.1
Ratio to Maxwellian TM=1.35 MeV
Ratio to Maxwellian TM = 1.35 MeV
1.4
1.0
0.9
0.8
exp IAEA/CRP Jan.2011
Bojkov
Starostov
0.7
0.6
PbP calculation
0.5
0
1
2
3
4
5
6
7
8
9
232
Th(n2 MeV,f)
1.0
0.8
Baba JPNKTO 1989
PbP calculation
0.6
0.4
232
1.2
Th(n2.9 MeV,f)
1.0
0.8
Bojkov RUSFEI 1991
PbP calculation
0.6
0.4
10 11 12 13 14 15
0
E (MeV)
1
2
3
4
5
6
7
8
E (MeV)
9
10 11 12 13 14 15
a
Fig.6: Examples of PFNS results (red line) a) U(nth,f) and b) 232Th(n,f) at En = 2 MeV and 2.9 MeV in
comparison with experimental data (different open and full symbols) plotted as ratio to a Maxwellian
spectrum.
233
9
b
3.0
3.0
Prompt neutron multiplicity
2.5
239
PbP calculation
3Z/A, Z, rms Wahl
Y(A,TKE) Straede et al.
U(nth,f)
2.5
Prompt neutron multiplicity
235
2.0
1.5
1.0
0.5
0.0
PbP calculation
3Z/A, Z, rms Wahl
Y(A,TKE) IRMM Wagemans
Pu(nth,f)
2.0
1.5
1.0
0.5
0.0
30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62
Z
32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62
a
Fig.7: Examples of ν(Z) results for a) 235U(nth,f) and b) 239Pu(nth,f).
10
Z
b