Benchmarking nuclear fission theory

The concept of a fission barrier height is fraught with ambiguity [3]. A theoretical definition is the energy difference between the ground state and the highest saddle point in a shape-constrained potential energy surface (PES) that has the lowest energy for all possible paths leading to fission from the ground state. If the theory treats the angular momentum of the nucleus, the benchmark should be for the PES corresponding to the angular momentum of the ground state. We have chosen 15 examples for the benchmarks, including the well-known nuclei for reactor physics, and some examples with isotope chains ranging from Z = 90 to Z = 96 and an example beyond Pu to better exhibit the Z-dependence of the barriers. The empirical barriers are taken from RIPL-3 compilation [4].

Contrary to cross sections, fission barriers are not physical observables, and 'empirical' barriers are inferred from measured cross sections using particular models for the PES, the collective inertia tensor, and the level density on top of the barrier. The presence of a double-humped, or more complicated, structure along the predicted fission pathways further complicate matters as significant deviations from the traditional Hauser–Feshbach calculations of fission probabilities have to be considered.

The study in [5] concludes that fission barrier heights can be known to about ±0.3 MeV, with only little sensitivity to the particular prescription chosen for describing the level density on top of the barrier. We should consider this uncertainty as a lower limit, since several simplifying assumptions were made in deriving the barrier heights from experimental cross sections/penetrabilities. More complicated fission pathways in a multi-dimensional potential energy landscape, multi-modal fission, non-parabolic fission barriers, etc, can all increase this uncertainty. A more reasonable estimate would be 0.5−1.0 MeV in most cases.

In addition to providing benchmark values in table 1 against which theoretical calculations can be compared, trends in inner and outer fission barrier heights as a function of mass number and fissility parameter ${{Z}^{2}}/A$ should also serve as a guide. For lower-Z actinides, e.g., Th isotopes, inner barrier heights are lower than outer barrier heights. This trend is reversed for heavier actinides, e.g., Cm isotopes.

One of the most challenging aspects of fission theory is to correctly predict the energies and half-lives of the superdeformed intermediate states of the fissioning nucleus, the spontaneously fissioning shape isomers . The excitation energies are typically 2–3 MeV in the second minimum of the fission barrier. Spectroscopic studies of the transitions between the states in the second minimum have shown that the moments of inertia associated with the rotational bands are those expected for nuclei with an axes ratio of 2:1—a result confirmed by studies of the quadrupole moments [6]. All of these facts represent a significant constraint on, and a challenge for, fission theories.

Table 2.  Table of (even–even) Fission isomer excitation energies ${{E}_{{\rm II}}}$ [8, 9].

Nuclide	${{E}_{{\rm II}}}$ (keV)	${{T}_{1/2}}$
236U	2750	120 ns
238U	2557.9	280 ns
238Pu	∼2400	0.6 ns
240Pu	∼2800	3.7 ns
242Pu	∼2000	28 ns
	2000	3.5 ns
240Cm	∼3000	55 ns
242Cm	∼1900	40 ps
244Cm	∼2200	$\leqslant 5$ ps

An isomer excitation energy can be obtained by analyzing experimental data on the excitation energy dependence of the cross sections for formation of the isomer, and in particular near the threshold of the rising curves. Most of these experimental data come from neutron evaporation and particle transfer reactions. As for fission barrier heights, the inferred isomer energy is model-dependent, and has to be considered carefully.

As discussed in [7], the analysis of the experimental excitation curves is easier in the case of fissioning doubly-odd nuclei, where simplifying assumptions can be made on the level density representation used in the cross section calculations. Even in those cases, however, the uncertainty on the isomer energy is probably at least equal to the uncertainty (∼0.5–1.0 MeV) on fission barrier heights, as discussed above.

The examples chosen in table 3 are for illustrative purposes only. Many more spontaneous fission half-lives have been measured and analyzed, as reported in [10]. For the examples we have chosen the well-known ²⁴⁰Pu lifetime together with two cases among heavier actinide elements that exhibit extreme variations in lifetimes. Some of these lifetimes were evaluated by calculating a weighted average of available experimental data, while others were based on selecting on particular experimental value only, deemed of higher quality by the evaluators. We included this information in table 3. In most cases, the most trustful values would be the ones obtained as a weighted average, as at least two consistent experimental data were included.

Table 3.  Spontaneous fission half-lives [10].

Nuclide	${{T}_{{\rm SF}}}$	Evaluation method
240Pu	1.14 ± 0.010 x 1011 years	Weighted average
252Cf	86 ± 1 year	Weighted average
254Fm	228 ± 1 day	Selected value
258Fm	0.37 ± 0.02 ms	Weighted average
256Rf	6.2 ± 0.2 ms	Weighted average
260Rf	20 ± 1 ms	Selected value

It is worth noting that when dealing with quantities that can vary by many orders of magnitude, it makes sense to compare not the differences between theory and experiment but rather the logarithm of the ratio of theory to experiment,

$$\begin{eqnarray}&&{{R}_{x}}={\rm log} \left( \frac{{{x}_{{\rm th}}}}{{{x}_{{\rm exp} }}} \right).\end{eqnarray} \tag{ 1 }$$

The target performance measures are then the mean value of R_x,

$$\begin{eqnarray}&&{{\bar{R}}_{x}}=\frac{1}{{{N}_{d}}}\mathop{\sum }\limits_{i}{{R}_{x,i}},\end{eqnarray} \tag{ 2 }$$

and the variance about the mean

$$\begin{eqnarray}&&\sigma =\frac{1}{{{N}_{d}}}{{\left( \mathop{\sum }\limits_{i}{{({{R}_{x,i}}-{{{\bar{R}}}_{x}})}^{2}} \right)}^{1/2}}.\end{eqnarray} \tag{ 3 }$$

Here N_d is the number of data points in the benchmark set. We note that these measures are in common use, for example in reporting the performance of theories of the nuclear level density [11]. Of course, if the model makes use of a parameter to fit benchmark data or data of the same kind, only the σ value provides an interesting test of the theory.

Fission fragment yields are commonly characterized by independent, cumulative and chain mass yields. Establishing meaningful benchmarks is complicated by the fact that there is no direct relation between what theories predict and what experiments measure.

Experimentally, the best-known mass yields are for the thermal neutron-induced fission reactions on ²³⁵U and ²³⁹Pu. Precise measurements (1−2%) have often been made using radiochemical techniques, in which cumulative yields are measured. Inferring the independent yields from those measurements therefore requires some modeling. Finally, fission theories will predict pre-neutron emission fission fragment yields, while experimental data always correspond to post-neutron emission yields. Here we limit the recommended fission fragment mass distribution benchmarks to only a few spontaneous fission or thermal neutron-induced fission reactions for which the average prompt neutron multiplicity as a function of the fragment mass is known relatively well.

However, for benchmarking purposes, we just recommend only two quantities that should be easier to compute and reflect the coarsest features of the distribution. We first determine the average mass A_m as

$$\begin{eqnarray}&&{{A}_{m}}=\frac{1}{P}\mathop{\sum }\limits_{A}AP(A),\end{eqnarray} \tag{ 4 }$$

where $P={{\sum }_{A}}P(A)$ is the total probability. Note that P = 1 is not precisely satisfied in the evaluated data tables. The experimental A_m comes out a few units less than half the mass number of the original nucleus, due to the emission of the prompt neutrons.

The benchmarks are the following two moments of the distribution for the higher mass fragments:

$$\begin{eqnarray}&&{{S}_{\gt }}=\frac{1}{{{P}_{\gt }}}\mathop{\sum }\limits_{A\gt {{A}_{m}}}P(A)\left( A-{{A}_{m}} \right),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\sigma _{\gt }^{2}=\frac{1}{{{P}_{\gt }}}\mathop{\sum }\limits_{A\gt {{A}_{m}}}P(A){{\left( A-{{A}_{m}} \right)}^{2}}-S_{\gt }^{2}.\end{eqnarray} \tag{ 6 }$$

Here ${{P}_{\gt }}$ is the total probability of producing fission fragments of mass higher than A_m:

$$\begin{eqnarray}&&{{P}_{\gt }}=\mathop{\sum }\limits_{A\gt {{A}_{m}}}P(A).\end{eqnarray} \tag{ 7 }$$

In simple models ${{P}_{\gt }}$ will be equal to one, but the experimental value differs from that by a small amount.

For the experimental cases, we include the thermal neutron-induced fission of ²³⁵U, ²³⁹Pu and ²⁵⁵Fm. The first two have the classic asymmetric mass yields and the latter has a more centered yield curve. Also we consider an example of spontaneous fission of ²⁵²Cf. The moments in table 4 were extracted from the experimental $P(A)$ data compiled in [12, 13]. The table also gives the values of A_m and ${{P}_{\gt }}$ for the data, although these are not part of the benchmark.

Table 4.  Fission product mass distribution characteristics extracted from the experimental data compiled in [12]. (The data are available in a tabulated text form in [13].)

Nuclide	Am	${{P}_{\gt }}$	${{S}_{\gt }}$	${{\sigma }_{\gt }}$
236U*	116.7	0.98	22.0	5.1
240Pu*	118.3	0.96	19.9	5.7
252Cf	124.0	0.99	18.0	6.4
256Fm*	126.4	0.97	12.3	6.9

^*denotes induced fission by thermal neutron capture on the $A-1$ isotope.

Table 5.  Recommended [14] average pre-neutron evaporation total kinetic energies of the fission fragments.

Reaction	$\langle {\rm TKE}\rangle$ (MeV)
233U (${{n}_{{\rm th}}},f$)	170.1 ± 0.5
235U (${{n}_{{\rm th}}},f$)	170.5 ± 0.5
239Pu (${{n}_{{\rm th}}},f$)	177.9 ± 0.5
252Cf (sf)	184.1 ± 1.3

The total kinetic energy (TKE) of the fission fragments is an important quantity for several reasons. It is an indicator for the shape of the fission fragments near their scission configurations: the higher the TKE value, the more compact the nascent fragments are. This quantity also directly influences the excitation energy left in the initial fragments, which is released through the evaporation of neutrons and photons. It also represents an important benchmark for fission theories to compute.

The average pre-neutron evaporation total kinetic energies $\langle {\rm TKE}\rangle$ for ²⁵²Cf spontaneous fission and thermal neutron-induced fission of $^{233,235}$U and ²³⁹Pu are considered as energy standards [14]. To a first-order, the evolution of $\langle {\rm TKE}\rangle$ follows the Coulomb parameter ${{Z}^{2}}/{{A}^{1/3}}$. The evolution of the average total kinetic energy of the fission fragments with the incident neutron energy was addressed in [15], by fitting available experimental data. The fitted values for the thermal energy point for ²³⁵U and ²³⁹Pu are consistent within one-sigma with the values recommended in [14].