Does Spontaneous Fission Siscrete or Continuous Energy

Nuclear physics and basic technology

British Electricity International , in Nuclear Power Generation (Third Edition), 1992

3.3.2 Number of neutrons emitted per fission (v)

In each fission event the number of neutrons emitted must be an integer. For a very large number of fissions of U-235 by thermal neutrons it is known that 2.7% of the fissions give no neutrons, 15.8% give one neutron, 33.9% give two, and so on. The average number of neutrons released per fission is denoted by v and has a value of 2.43 for thermal fission in U-235.

The average number of neutrons released per fission is different for different fissile isotopes. Also, v increases more or less linearly with increasing neutron energy giving an extra neutron for each 7 MeV of neutron kinetic energy.

Table 1.3 gives the values of v for the fission of uranium and plutonium by thermal and 1 MeV neutrons (or the threshold 1.1 MeV for U-238).

TABLE 1.3. Number of neutrons emitted per fission (v) of U-235, Pu-239, U-238

Isotope	Incident neutron energy	v
U-235	0.025 eV	2.43
	1.0 MeV	2.50
Pu-239	0.025 eV	2.89
	1.0 MeV	3.02
U-238	0.025 eV	0
	1.1 MeV	2.46

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Advanced Fuels/Fuel Cladding/Nuclear Fuel Performance Modeling and Simulation

P. Van Uffelen , M. Suzuki , in Comprehensive Nuclear Materials, 2012

3.19.2.3.1.1 Recoil, knockout, and sputtering

In general, a fission event entails – among others – two fission fragments that convey their kinetic energy to the fuel lattice. A fission fragment close enough to a free surface (<6–7  μm) can escape from the fuel due to its high kinetic energy (60–100   MeV). This is called recoil release. When fission fragments make elastic collisions with the nuclei of the lattice atoms, a collision cascade begins. The interaction of a fission fragment, a collision cascade, or a fission spike with a stationary gas atom near the surface can also cause the latter to be ejected if it happens within a distance close enough to the surface. This process is called 'release by knockout.' Finally, a fission fragment traveling through oxide loses energy, causing a high local heat pulse. When this happens close to the fuel surface, a heated zone will evaporate or sputter, thereby releasing any fission product contained in the evaporated zone.

Recoil, knockout, and sputtering can only be observed at temperatures below 1000   °C, when thermally activated processes (see Sections 3.19.2.3.1.2 , 3.19.2.3.1.3 , 3.19.2.3.1.5 , 3.19.2.3.1.6 , 3.19.2.3.1.7 , 3.19.2.3.1.8 ) do not dominate. They are almost temperature independent and therefore called 'athermal mechanisms.' It is generally of little importance in a reactor at intermediate burnup levels. The fraction of athermal release is roughly under 1% for rod burnups below 45   MWd   kgU⁻¹, and increases to roughly 3% when the burnup reaches about 60   MWd   kgU⁻¹.

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Nonproliferation and safeguards aspects of the MSR fuel cycle

Sophie Grape , Carl Hellesen , in Molten Salt Reactors and Thorium Energy, 2017

10.3.2 Spontaneous neutron generation

Neutrons are released from spontaneous fission events in the weapons material. If such a neutron is generated during the implosion phase of an NED, but before it has reached optimum reactivity, a chain reaction can be initiated prematurely. Some nuclear weapon designs are sensitive to preinitiation, and the use of materials with a comparatively strong neutron generation, such as reactor-grade plutonium, can substantially degrade the yield of such designs ( DOE, 1997). Therefore, for an unadvanced proliferator, preinitiation is an issue if a high and reliable yield is desired. However, it should be mentioned in this context that even if preinitiation occurs at the worst possible moment, when the material first reaches prompt criticality, the explosive yield of a relatively simple first-generation device will still be of the order of one or a few thousand tons of TNT equivalent. This is still a significant yield and far greater than any conventional weapon ever manufactured.

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Criticality analysis of packages for radioactive materials

C.V. Parks , in Safe and Secure Transport and Storage of Radioactive Materials, 2015

10.1 Introduction

A self-sustaining chain of fission events is termed a 'nuclear criticality'. Thus, when a system is critical, the generation of neutrons is equal to the loss of neutrons and the effective neutron multiplication factor ( k _eff) is 1.0. The goal of criticality safety is to assure subcriticality – that is, k _eff  <   1.0 – for all conditions that a system outside of a reactor might encounter. Radioactive material containing fissile nuclides is typically termed fissile material and criticality safety during the transport of such material must be ensured in order to protect workers and the public from the radiological effects of an unanticipated nuclear criticality event. Fissile nuclides are defined to be U-233, U-235, Pu-239 and Pu-241. The regulations governing the transport of fissile material are established to provide requirements for package design and performance and operational controls that must be adhered to during transport. A criticality safety assessment is the term used to describe the process that must be undertaken to demonstrate that subcriticality will be maintained under both normal conditions of transport (NCT) and accident conditions of transport (ACT). This chapter will provide an overview of the regulatory requirements for design and transport of packages that contain fissile material, address methodologies and approaches used to perform the criticality safety assessment, and discuss the complexities and challenges related to reliable assurance of criticality safety.

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The Fission Process

Malcolm Joyce , in Nuclear Engineering, 2018

4.6 Different Modes of Fission

If the more massive isotopes have a predisposition toward fission, this begs the question as to why some of these isotopes fission spontaneously where as others require a stimulus to do so; indeed why is it that the heavy nuclei beyond a limiting mass do not just fission leaving nothing else behind? The answer to these questions is complicated but our understanding of it benefits from a closer consideration of the energy that is needed to overcome the cohesion of the nucleus. The cohesion constitutes a threshold commonly referred to as the activation energy; this is the energy necessary to overcome what is termed the Coulomb barrier.

It is useful in this context if we think of a fission event happening in reverse, albeit hypothetically. In this scenario, as the two charged fragments are brought closer together, the Coulomb potential between them increases (along with the force of repulsion) as the separation r gets smaller. This is illustrated by the expression for Coulomb potential, V, and its inverse proportionality to the separation r of the fragments of atomic number Z_a and Z_b , in Eq. (4.13),

(4.13) $V=\frac{1}{4\pi {\varepsilon }_0}\frac{Z_a{Z}_b{e}^2}{r}$

where e is the charge on the electron and ɛ ₀ is the relative permittivity of free space, 8.85   ×   10⁻¹²  Fm^{−   1}. The potential energy increases due to the inverse proportionality with separation until the parent nucleus coalesces. The binding energy of this nucleus is represented by a well that is incorporated into the trend with r for $r<r_0$ depicted as per Fig. 4.10 where r ₀ is the radius of the coalesced nucleus. Based on this illustration, three scenarios are possible:

Fig. 4.10. A schematic depiction of energy versus separation of the fragmentation of a nucleus in fission.

1.

A nucleus might possess sufficient energy to place it above the Coulomb barrier. In this case, the activation energy is zero and there is nothing to prevent the nucleus proceeding immediately to fission, notwithstanding the influence of any quantum mechanical effects of the energy level structure of individual nucleons, which we shall ignore for the purposes of this illustration. For the very massive isotopes that might exist in this category, that is, A  >   300, none exist for sufficiently long to have been identified, at least by current methods, due to their instability toward spontaneous fission.

2.

A nucleus can have an excitation energy that places it near to the top of the Coulomb barrier but it does not have sufficient energy to pass over it. However, as a subatomic system behaving according to quantum mechanics, there is an appreciable probability that it might tunnel through the barrier to fission. This is the mechanism behind spontaneous fission, such as is observed in the case of ²⁵²Cf, ²⁴⁴Cm, ²⁴⁰Pu and ²³⁸U, for example. Generally speaking, the level of excitation energy and thus the relative thickness of the Coulomb barrier has a significant influence on the probability of fission and consequently the observed half-life.

3.

Or a nucleus might reside in the potential well with an excitation energy that is too low to yield any noticeable level of spontaneous fission because the probability of tunnelling through the relatively thick barrier at this level is too small. However, the absorption of a neutron (or photon) might place the nucleus so formed high relative to the barrier and thus cause it to be advantageously predisposed to fission. Hence stimulated fission is observed as in the case of ²³⁵U and ²³⁹Pu when these isotopes are effectively lifted by the absorption of a thermal neutron to the excited forms of ²³⁶U⁎ and ²⁴⁰Pu⁎, respectively. For the case of those isotopes that are not vulnerable to stimulated fission with thermal neutrons, such as ²³⁸U, the probability of fission can be raised by stimulation with higher energy neutrons. This is termed above-threshold fission; the higher incident energy is sufficient to yield the intermediate nucleus formed as a result of the neutron absorption to be higher in energy where the thickness of the barrier to fission is reduced relative to that at thermal energies.

While it is instructive to think of the parent nucleus as comprising two fragments preformed within the potential, this is unlikely to be the case in reality. Nonetheless, this does not undermine the benefit of the treatment based on the Coulomb barrier described above to understanding this complicated phenomenon; it is similar to the widely accepted theoretical treatment of α decay, with the α particle considered to be in some way preformed. The barrier methodology also enables half-lives to be predicted reasonably well and it supports the concept of activation energy, although this analysis is beyond the scope of this text. It also assists our understanding of the distinction between the spontaneous and stimulated fission decay pathways. Thus, the predisposition of matter to fission can be considered in terms of those isotopes susceptible to spontaneous fission, and those susceptible to stimulated fission; the former exhibit significant range in half-life but fission often only accounts for a very small fraction of the total decay probability of the isotope with the majority usually being confined to other modes of disintegration, principally α decay. Stimulated fission (also known as induced fission) affords the opportunity to establish and control a fission chain reaction. A discussion of this, and its industrial use, forms the focus of the subsequent chapters of this text.

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NEUTRON ATTENUATION

JAMES WOOD , in Computational Methods in Reactor Shielding, 1982

Source

As was shown in Example 3.6 , if we assume a uniform distribution of fission events throughout the core, the fission rate per unit volume of the core is

(6.43) $\begin{array}{cc}{\text{S}}_{\text{V}}^{\text{f}}=3\cdot 1\times {10}^{16}\frac{\text{P}}{\text{V}}& \text{fission}\end{array}{\text{cm}}^{-3}{\text{s}}^{-1}.$

If we make the further simplification that the primary photons are emitted with a number of discrete energies E, and associated weights, η_c, then the primary gamma volume source for photons of energy E has strength,

(6.44) $\begin{array}{cc}{\text{S}}_{\text{V}}^{\text{g}}=3\cdot 1\times {10}^{16}{\text{n}}_{\text{C}}\left(\text{E}\right)\frac{\text{P}}{\text{V}}& \text{photon}\end{array}{\text{cm}}^{-3}{\text{\hspace{0.17em}s}}^{-1},$

where η_c(E) is the number of photons emitted per fission into the energy band designated by E.

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Nuclear reactor kinetics

A. John Arul , ... Om Pal Singh , in Physics of Nuclear Reactors, 2021

7.3.2 Delayed photo neutrons

As indicated in the previous paragraph, delayed neutrons may be emitted following a fission event through an alternate route, which involves a photo nuclear reaction. In a photo nuclear reaction an incident gamma ray photon causes a neutron to be emitted from a light nucleus. For this reaction to occur the energy of the gamma ray photon has to be greater than the binding energy of the neutron in that nucleus. The binding energy of some important light nuclei is given in Table 7.5.

Table 7.5. Some important photo nuclear reactions and their threshold energies [11].

Nuclide	Reaction	Threshold energy (MeV)
2H	2H(γ,n)1H	2.225
6Li	6Li(γ,n)5Li	5.67
6Li	6Li(γ,α)4He	3.697
7Li	7Li(γ,n)6Li	7.251
9Be	9Be(γ,n)8Be	1.667
13C	13C(γ,n)12C	4.9

This process is analogous to delayed neutron emission discussed earlier. But instead of the daughter nucleus of the precursor, left in the excited state directly emitting neutrons, the gamma rays emitted by the nucleus interacts with light nuclei such as deuterium or lithium to give rise to a neutron. The delay is due to the beta decay as in the case of delayed neutrons. The photo-delayed neutrons produced by this process constitute a relatively small but significant fraction of delayed neutrons. In an Indian pressurized heavy water reactor (PHWR) or a CANDU reactor the delayed photo neutron fraction is about 0.13 mk from U-235 fissions [9]. For comparison, the delayed neutron fraction from precursor decay is about 5 mk. Even in a light water reactor (LWR), there is some amount of delayed photo neutrons produced as the LWR consists of about 0.015% D₂O. Therefore for thermal reactor kinetics calculations, the delayed photo neutrons are treated as additional groups (9–11 groups) of delayed neutrons with respective decay constants and yields [8, 11, 12]. While the direct delayed neutron group has a maximum half-life of approximately 1   min, the delayed photo neutrons have a maximum of half-life of approximately 13   days (¹⁴⁰Ba). In PHWR the photo-neutrons associated with the very long half-life fission products are useful to provide sufficient neutron flux for startup monitoring. Some of the fission product beta decay chains, which involve high energy gamma emissions, are given in the following Table 7.6 along with their half-lives [11]. Photo neutrons are produced mainly by decomposition of D (into a neutron and a proton) with incident gamma of energy greater than binding energy of D (2.3 MeV).

Table 7.6. High energy γ-emitting long life fission products.

Nuclide	Half life
140Ba-140La-140Ce	12.75   days, 1.68   days
132Te-132I	3.2   days, 0.956   days
106Ru-106Rh	374   days, 29.8   s
156Eu	15.2   days

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Nuclear reactors

Philip Thomas , in Simulation of Industrial Processes for Control Engineers, 1999

21.8 The balance for delayed neutron precursors

Let the production of the nuclei of delayed neutron precursor i due to absorption in a fission event be δC_fi . We may see from Figure 21.3 that k_dn_G delayed neutron precursor nuclei per m³ are produced at every absorption event, which events occur every δt seconds. It follows that

(21.25) $\begin{array}{ll}\delta C_{fi}=k_{di}{n}_G\hfill & i=1\text{to}6\hfill \end{array}$

Following the procedure of Section 21.6, we may substitute n_G = (n/l)δt and let δt → 0 to give the rate of production of delayed neutron precursor nuclei per m³ as

(21.26) $\begin{array}{ll}\frac{d{C}_{fi}}{dt}=k_{di}\frac{n}{l}=k{\beta }_i\frac{n}{l}\hfill & i=1\text{to}6\hfill \end{array}$

where equation (21.6) has been used in the last step.

The rate of increase of concentration of delayed neutron precursors is the rate of production minus the rate of decay:

(21.27) $\begin{array}{ll}\frac{d{C}_i}{dt}=\frac{d{C}_{fi}}{dt}-R_i\hfill & i=1\text{to}6\hfill \end{array}$

and using equations (21.1) and (21.26), we may re-express this as:

(21.28) $\begin{array}{ll}\frac{d{C}_i}{dt}=\frac{k{\beta }_i}{l}n-{\lambda }_i{C}_i\hfill & i=1\text{to}6\hfill \end{array}$

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Radiation Sources

Richard E. Faw , J.Kenneth Shultis , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

IV.B Prompt Fission Photons

The fission process produces copious gamma photons either within the first 6   ×   10⁻⁸ sec after the fission event (the prompt fission gamma photons) or from the subsequent decay of the fission products. These photons are of extreme importance in the shielding and gamma-heating calculations for a nuclear reactor. Consequently, much effort has been directed toward determining their nature.

Most investigations of prompt fission gamma photons have centered on the thermal-neutron-induced fission of ²³⁵U. For this nuclide it has been found that the number of prompt fission photons is 8.13   ±   0.35 photons per fission over the energy range 0.1 to 10.5   MeV, and the energy carried by this number of photons is 7.25   ±   0.26   MeV per fission. The energy spectrum of prompt gamma photons from the thermal fission of ²³⁵U between 0.1 and 0.6   MeV is approximately constant at 6.6 photons MeV⁻¹ fission⁻¹. At higher energies the spectrum falls off sharply with increasing energy. The measured energy distribution of the prompt fission photons can be represented by the following empirical fit over the range 0.1 to 10.5   MeV:

$\text{N}\left(\text{E}\right)=\left\{\begin{array}{cc}\hfill 6.6& \hfill 0.1<\text{E}<0.6\text{Mev}\hfill \\ \hfill 20.2{\text{e}}^{-1.78\text{E}}& \hfill 0.6<\text{E}<1.5\text{Mev}\hfill \\ \hfill 7.2{\text{e}}^{-1.09\text{E}}& \hfill 1.5<\text{E}<10.5\text{Mev},\hfill \end{array}\right)$

where E is in MeV and N(E) is in units of photons MeV⁻¹ fission⁻¹.

Investigation of ²³³U, ²³⁹Pu, and ²⁵²Cf indicates that the prompt fission photon energy spectra for these isotopes resembles very closely that for ²³⁵U, and hence for most purposes it is reasonable to use the ²³⁵U spectrum for other fissioning isotopes.

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Neutron Flux Energy Distribution

William EmrichJr., in Principles of Nuclear Rocket Propulsion, 2016

3 Energy Distribution of Neutrons in the Fission Source Range

The fission source range is defined as that range of neutron energies within which neutrons appear as a result of fission events. Generally speaking, this energy range extends from about 10  keV to about 10   MeV and corresponds to the χ(E) distribution discussed earlier. If one neglects scattering in this energy range, the neutron balance equation can be written as:

$\text{Removal}\text{Rate}=\text{Production}\text{Rate}$

The above-mentioned neutron balance equation may also be written as:

(6.30) ${\text{Σ}}_{\text{t}}\left(E\right)\varphi \left(E\right)=\chi \left(E\right){\int }_0^{\infty }\nu \left(E^{\prime }\right){\text{Σ}}_{\text{f}}\left(E^{\prime }\right)\varphi \left(E^{\prime }\right)d{E}^{\prime }$

where ν(E′) is the number of neutrons emitted on average per fission. This function is actually a very weak function of energy and is, therefore, often treated as a constant. In Eq. (6.30) the integral evaluates to a constant, therefore:

(6.31) ${\text{Σ}}_{\text{t}}\left(E\right)\varphi \left(E\right)=C\chi \left(E\right)$

Using Eq. (6.31), the neutron flux may now be written as:

(6.32) $\varphi \left(E\right)=\frac{C\chi \left(E\right)}{{\text{Σ}}_{\text{t}}\left(E\right)}$

Since Σ_t(E) is fairly constant at high energies, the neutron flux; therefore, approximately follows the χ(E) distribution in the fission source energy range.

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