Outer electron orbital
2.3 Correlation between particles from fission
2.3
Correlation between particles from fission
1150
As described in the previous sections, following spontaneous and induced fission, a number
1151
of prompt neutrons andγ rays are emitted [34, 35, 50, 51, 52]. All particles released during a
1152
fission event are correlated3 to each other in four domains: i) number of particles released; ii) 1153
temporal separation between the released particles; iii) spatial separation between the released
1154
particles; and iv) the energies at which the particles are emitted. Such correlations have been
1155
studied widely [36, 37, 38].
1156
The neutronnumber distributions (i.e. probability distribution functions outlining the likeli-
1157
hood of a given number of neutrons,n, that may be emitted following fission) of some common
1158
spontaneously fissile isotopes are illustrated in figure 2.8(a) and table 2.2(a). These number
1159
distributions depend on the mass of the fission fragments that are created during the fission
1160
process [54], which in turn is dependent on the mass of the parent isotope and the excitation
1161
energy of the inducing neutron (latter is valid for induced fission only). Such correlation may
1162
also be noticed in the promptγrays that are emitted during spontaneous fission [55], as shown
1163
in figure 2.8(b).
1164
Further to this, each of the prompt neutrons and γ rays expelled from the parent nucleus
1165
have different times of emission but are clustered together in the sub-nanosecond region (i.e. <
1166
10−13 second [44]). Additionally, as the fission fragments break away, the energies with which
1167
they escape are correlated to one another [56]. As the subsequent particles that are emitted
1168
share among themselves the energy that the fission fragments gained during the fission process,
1169
this gives rise to the energy correlation between them. This is not to be confused with the
1170
Maxwellian statistical distribution, which is widely used for the energy distribution of the average
1171
or individual neutrons that are emitted from a fission isotope. Here, correlation refers to the fact
1172
that the energy of the first neutron, which itself has a Maxwellian statistical distribution, will
1173
impact the energy of subsequent neutrons, i.e. their position in the Maxwellian distribution.
1174
A significant proportion of the neutrons expelled during spontaneous and induced fission are
1175
emitted from two fission fragments which usually have unequal mass. These fragments move away
1176
from each other due to the kinetic energy gained during the fission process. Since 95% of all the
1177
particles emitted during the fission process are from fully accelerated fragments [48], the released
1178
particles contain part of that momentum in accordance with conservation law. As a consequence,
1179
neutrons emitted from a single fission fragment will be polarized in the same direction (i.e. the
1180
emitted neutrons will have a small angular separation); whereas neutrons emitted from two
1181
complementary fragments will be focused in opposing directions (i.e. the emitted neutrons will
1182
3A mutual relationship or connection, i.e. interdependence, between two or more things, e.g. the energy of
(a) Neutron number distribution for spontaneous fission of various isotopes.
(b)γ-ray number distribution for spontaneous fission of various isotopes.
Figure 2.8|Neutron andγ-ray number distributions following spontaneous fission of various isotopes. Illustration of the(a)neutron and(b)γ-raynumber distributionsfollowing spontaneous fission of various isotopes. These data points are discrete distributions and the straight-line fit was added to guide the eye only. The distributions were obtained from the FREYA libraries [53] using a C++ script (see appendix D.1).
2.3. Correlation between particles from fission 27
Figure 2.9 | Angular correlation of neutron and γ-ray particles from spontaneous fission of252Cf. Angular separations between the particles emitted from the spontaneous fission
of252Cf isotope extracted from the FREYA library [53] using a C++ script (see appendix D.2).
have a large angular separation). Thus, the neutrons originating from fissioning isotopes will
1183
have an anisotropic spatial correlation, i.e. they are emitted preferentially near 0 and π rad
1184
relative to each other. Additionally, the rotation of the fission fragments is also documented to
1185
have a small influence on the anisotropy of the distribution [48, 57]. The number of neutrons
1186
that are emitted during the descent from saddle to scission and during the acceleration of the
1187
fragments is limited, as only 5% of the emitted neutrons fall in this category, but may still have a
1188
discernible contribution towards the spatial anisotropy. These trends in spatial distribution are
1189
illustrated in figure 2.9 for252Cf. 1190
2.3.1
Fission models for correlated particles
1191
There are several models that have evolved over the past decades which can be used to predict
1192
the characteristics of neutrons andγ rays that are emitted from fission events [48, 59]. These
1193
include, but are not limited to:
1194
1. CGMF which is an implementation of the statistical Hauser-Feshbach nuclear reaction
1195
theory [60] applied to the de-excitation of the primary fission fragments which are described
1196
as compound nuclei with an initial excitation energy, spin and parity. Each emitted neutron
1197
andγ-ray particle removes its kinetic energy from the fragment’s intrinsic excitation energy,
1198
while doing little to change the fragment’s angular momentum [61, 62].
1199
2. Fission Reaction Event Yield Algorithm (FREYA)which generates complete fission events
providing the full kinematic information on the fission products, and all the subsequently
1201
emitted neutrons and photons, by relying on experimental data; and is supplemented using
1202
a simple physics-based model when no experimental data are available [53, 57].
1203
3. FIFRELIN which is based on empirical models associated with macroscopic or microscopic
1204
ingredients or both with the fission fragment de-excitation being performed within Weis-
1205
skopf (for uncoupled neutron andγ-ray emission) or Hauser-Feshbach (for coupled neutron
1206
andγ-ray emission) statistical theory [63].
1207
To complete this thesis, the FREYA model [53, 57, 64] was used for modelling correlated
1208
particles. It uses experimental data for neutron and γ-ray4 number distributions (i.e. P ν for 1209
neutron andGforγray) from spontaneous fission (see table 2.2(a) on page 24). If no data exist, it
1210
uses Terrell’s approximation [65] in equation 2.20 for neutron and Valentine’s approximation [66]
1211
in equation 2.21 forγ-ray emissions, with parameters taken from Ensslin [34].
1212 ν X n=0 Pn = 1 √ 2π Z ν−ν¯+0σ.05+b −∞ exp−t 2 2 dt (2.20) 1213 Y (G) =a+G+ 1 G a a+ ¯G G 1− a a+ ¯G , wherea≈26 (2.21) 1214 En= r πb 4a exp4ba a exp −cE0sinh(√bE0) (2.22)
The energy distributions of neutrons (En) from spontaneous fission events are defined using 1215
the Watt spectrum equation (see equation 2.22). The values of the coefficients of the Watt
1216
spectrum equation are taken from Ensslin [34] (see table 2.2(a) on page 24). For neutron-induced
1217
fission, FREYA uses TART’s implementation [67]. The energy correlation is then computed by
1218
the FREYA model by imposing a constraint on the total event energy of all emitted particles
1219
using a technique whereby the average outgoing promptγ-ray energy and prompt neutron energy
1220
are expressed by an actinide-dependent quadratic expression. In this method, the description of
1221
γ-ray spectra is limited to232U,235U,238U, 239Pu and 252Cf, whilst the neutron energy spectra
1222
is available for 73 different actinides based on Evaluated Nuclear Data Library 2008.
1223
4Experimental data forγnumber distributions are only available for spontaneous fission of252Cf. Others are