Periodic Table

• Mean square radius of a neutron is ~ 0.8 x 10^-15m (0.8 fermi)
• The mass of the neutron is 939.565 MeV/c²
• Neutrons are ½ spin particles – fermionic statistics
• Neutrons are neutral particles – no net electric charge.
• Neutrons have non-zero magnetic moment.
• Free neutrons (outside a nucleus) are unstable and decay via beta decay. The decay of the neutron involves the weak interaction and is associated with a quark transformation (a down quark is converted to an up quark).
• Mean lifetime of a free neutron is 882 seconds (i.e. half-life is 611 seconds ).
• A natural neutron background of free neutrons exists everywhere on Earth and it is caused by muons produced in the atmosphere, where high energy cosmic rays collide with particles of Earth’s atmosphere.
• Neutrons cannot directly cause ionization. Neutrons ionize matter only indirectly.
• Neutrons can travel hundreds of feet in air without any interaction. Neutron radiation is highly penetrating.
• Neutrons trigger the nuclear fission.
• The fission process produces free neutrons (2 or 3).
• Thermal or cold neutrons have the wavelengths similar to atomic spacings. They can be used in neutron diffraction experiments to determine the atomic and/or magnetic structure of a material.

See also:

Structure of the Neutron

See also:

Neutron

See also:

Neutron Energy

Classification of free neutrons according to kinetic energies

• Cold Neutrons (0 eV; 0.025 eV). Neutrons in thermal equilibrium with very cold surroundings such as liquid deuterium. This spectrum is used for neutron scattering experiments.
• Thermal Neutrons. Neutrons in thermal equilibrium with a surrounding medium. Most probable energy at 20°C (68°F) for Maxwellian distribution is 0.025 eV (~2 km/s). This part of neutron’s energy spectrum constitutes most important part of spectrum in thermal reactors.
• Epithermal Neutrons (0.025 eV; 0.4 eV). Neutrons of kinetic energy greater than thermal. Some of reactor designs operates with epithermal neutron’s spectrum. This design allows to reach higher fuel breeding ratio than in thermal reactors.

• 

  Cadmium Neutrons (0.4 eV; 0.5 eV). Neutrons of kinetic energy below the cadmium cut-off energy. One cadmium isotope, ¹¹³Cd, absorbs neutrons strongly only if they are below ~0.5 eV (cadmium cut-off energy).

• Epicadmium Neutrons (0.5 eV; 1 eV). Neutrons of kinetic energy above the cadmium cut-off energy. These neutrons are not absorbed by cadmium.
• Slow Neutrons (1 eV; 10 eV).
• Resonance Neutrons (10 eV; 300 eV). The resonance neutrons are called resonance for their special bahavior. At resonance energies the cross-sections can reach peaks more than 100x higher as the base value of cross-section. At this energies the neutron capture significantly exceeds a probability of fission. Therefore it is very important (for thermal reactors) to quickly overcome this range of energy and operate the reactor with thermal neutrons resulting in increase of probability of fission.
• Intermediate Neutrons (300 eV; 1 MeV).
• Fast Neutrons (1 MeV; 20 MeV). Neutrons of kinetic energy greater than 1 MeV (~15 000 km/s) are usually named fission neutrons. These neutrons are produced by nuclear processes such as nuclear fission or (ɑ,n) reactions. The fission neutrons have a mean energy (for ²³⁵U fission) of 2 MeV. Inside a nuclear reactor the fast neutrons are slowed down to the thermal energies via a process called neutron moderation.
• Relativistic Neutrons (20 MeV; ->)

The reactor physics does not need this fine division of neutron energies. The neutrons can be roughly (for purposes of reactor physics) divided into three energy ranges:

• Thermal neutrons (0.025 eV – 1 eV).
• Resonance neutrons (1 eV – 1 keV).
• Fast neutrons (1 keV – 10 MeV).

Even most of reactor computing codes use only two neutron energy groups:

• Slow neutrons group (0.025 eV – 1 keV).
• Fast neutrons group (1 keV – 10 MeV).

See also:

Properties of the Neutron

See also:

Neutron

See also:

Interactions of Neutrons with Matter

A free neutron is a neutron that is not bounded in a nucleus. The free neutron is, unlike a bounded neutron, subject to radioactive beta decay.

It decays into a proton, an electron, and an antineutrino (the antimatter counterpart of the neutrino, a particle with no charge and little or no mass). A free neutron will decay with a half-life of about 611 seconds (10.3 minutes). This decay involves the weak interaction and is associated with a quark transformation (a down quark is converted to an up quark). The decay of the neutron is a good example of the observations which led to the discovery of the neutrino. Because it decays in this manner, the neutron does not exist in nature in its free state, except among other highly energetic particles in cosmic rays. Since free neutrons are electrically neutral, they pass through the electrical fields within atoms without any interaction and they are interacting with matter almost exclusively through relatively rare collisions with atomic nuclei.

See also:

Interactions of Neutrons with Matter

See also:

Neutron

See also:

Application of Neutrons

Nuclear Reactors

A nuclear reactor is a key device of nuclear power plants, nuclear research facilities or nuclear propelled ships. Main purpose of the nuclear reactor is to initiate and control a sustained nuclear chain reaction. The nuclear chain reaction is initiated, sustained and controlled just via the free neutrons. The term chain means that one single nuclear reaction (neutron induced fission) causes an average of one or more subsequent nuclear reactions, thus leading to the possibility of a self-propagating series of these reactions. The “one or more” is the key parameter of reactor physics. To raise or lower the power, the amount of reactions, respectively the amount of the free neutrons in the nuclear core must be changed (using the control rods).

Neutron diffraction

Neutron diffraction experiments use an elastic neutron scattering to determine the atomic (or magnetic) structure of a material. The neutron diffraction is based the fact that thermal or cold neutrons have the wavelengths similar to atomic spacings. An examined sample (crystalline solids, gasses, liquids or amorphous materials) must be placed in a neutron beam of thermal (0.025 eV) or cold (neutrons in thermal equilibrium with very cold surroundings such as liquid deuterium) neutrons to obtain a diffraction pattern that provides information about the structure of the examined material. The neutron diffraction experiments are similar to X-ray diffraction experiments, but neutrons interact with matter differently. Photons (X-rays) interact primarily with the electrons surrounding (atomic electron cloud) a nucleus, but neutrons interact only with nuclei. Neither the electrons surrounding (atomic electron cloud) a nucleus nor the electric field caused by a positively charged nucleus affect a neutron’s flight. Due to their different properties, both methods together (neutron diffraction and X-ray diffraction) can provide complementary information about the structure of the material.

Applications in Medicine

Medical applications of neutrons began soon after the discovery of this particle in 1932. Neutrons are highly penetrating matter and ionizing, so they can be used in medical therapies such as radiation therapy or boron capture therapy. Unfortunately neutrons, when they are absorbed in matter, active the matter and leave the matter (target area) radioactive.

Neutron activation analysis

Neutron activation analysis is a method for determining the composition of examined material. This method was discovered in 1936 and stands at the forefront of methods used for quantitative material analysis of major, minor, trace, and rare elements. This method is based on neutron activation, where an analyzed sample is first irradiated with neutrons to produce specific radionuclides. The radioactive decay of these produced radionuclides is specific for each element (nuclide). Each nuclide emits the characterictic gamma rays which are measured using gamma spectroscopy, where gamma rays detected at a particular energy are indicative of a specific radionuclide and determine concentrations of the elements. Main advantage of this method is that neutrons does not destroy the sample. This method can be also used for determine an enrichment of nuclear material.

See also:

Free Neutron

See also:

Neutron

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Detection of Neutrons

Detection of neutrons is very specific, since the neutrons are electrically neutral particles, thus they are mainly subject to strong nuclear forces but not to electric forces. Therefore neutrons are not directly ionizing and they have usually to be converted into charged particles before they can be detected. Generally every type of neutron detector must be equipped with converter (to convert neutron radiation to common detectable radiation) and one of the conventional radiation detectors (scintillation detector, gaseous detector, semiconductor detector, etc.).

Neutron Converters

Two basic types of neutron interactions with matter are for this purpose available:

• Elastic scattering. The free neutron can be scattered by a nucleus, transferring some of its kinetic energy to the nucleus. If the neutron has enough energy to scatter off nuclei the recoiling nucleus ionizes the material surrounding the converter. In fact, only hydrogen and helium nuclei are light enough for practical application. Charge produced in this way can be collected by the conventional detector to produce a detected signal. Neutrons can transfer more energy to light nuclei. This method is appropriate for detecting fast neutrons (fast neutrons do not have high cross-section for absorption) allowing detection of fast neutrons without a moderator.
• Neutron absorption. This is a common method allowing detection of neutrons of entire energy spectrum. This method is is based on variety of absorption reactions (radiation capture, nuclear fission, rearrangement reactions, etc.). The neutron is here absorbed by target material (converter) emitting secondary particles such as protons, alpha particles, beta particles, photons (gamma rays) or fission fragments. Some reactions are threshold reactions (requiring a minimum energy of neutrons), but most of reactions occurs at epithermal and thermal energies. That means the moderation of fast neutrons is required leading in poor energy information of the neutrons. Most common nuclei for the neutron converter material are:
  • ¹⁰B(n,α). Where the neutron capture cross-section for thermal neutrons is σ = 3820 barns and the natural boron has abundance of ¹⁰B 19,8%.
  • ³He(n,p). Where the neutron capture cross-section for thermal neutrons is σ = 5350 barns and the natural helium has abundance of ³He 0.014%.
  • ⁶Li(n,α). Where the neutron capture cross-section for thermal neutrons is σ = 925 barns and the natural lithium has abundance of ⁶Li 7,4%.
  • ¹¹³Cd(n,ɣ). Where the neutron capture cross-section for thermal neutrons is σ = 20820 barns and the natural cadmium has abundance of ¹¹³Cd 12,2%.
  • ²³⁵U(n,fission). Where the fission cross-section for thermal neutrons is σ = 585 barns and the natural uranium has abundance of ²³⁵U 0.711%. Uranium as a converter produces fission fragments which are heavy charged particles. This have significant advantage. The heavy charged particles (fission fragments) create a high output signal, because the fragments deposit a large amount of energy in a detector sensitive volume. This allows an easy discrimination of the background radiation (e.i. gamma radiation). This important feature can be used for example in a nuclear reactor power measurement, where the neutron field is accompanied  by a significant gamma background.

Detection of Thermal Neutrons

Thermal neutrons are neutrons in thermal equilibrium with a surrounding medium of temperature 290K (17 °C or 62 °F). Most probable energy at 17°C (62°F) for Maxwellian distribution is 0.025 eV (~2 km/s). This part of neutron’s energy spectrum constitutes most important part of spectrum in thermal reactors.

Thermal neutrons have a different and often much larger effective neutron absorption cross-section (fission or radiative capture) for a given nuclide than fast neutrons.

In general, there are many detection principles and many types of detectors. In nuclear reactors, gaseous ionization detectors are the most common, since they are very efficient, reliable and cover a wide range of neutron flux. Various types of gaseous ionization detectors constitute so called the excore nuclear instrumentation system (NIS). The excore nuclear instrumentation system monitors the power level of the reactor by detecting neutron leakage from the reactor core.

Detection of Neutrons using Ionization Chamber

Ionization chambers are often used as the charged particle detection device. For example, if the inner surface of the ionization chamber is coated with a thin coat of boron, the (n,alpha) reaction can take place. Most of (n,alpha) reactions of thermal neutrons are 10B(n,alpha)7Li reactions accompanied by 0.48 MeV gamma emission.

Moreover, isotope boron-10 has high (n,alpha) reaction cross-section along the entire neutron energy spectrum. The alpha particle causes ionization within the chamber, and ejected electrons cause further secondary ionizations.

Another method for detecting neutrons using an ionization chamber is to use the gas boron trifluoride (BF₃) instead of air in the chamber. The incoming neutrons produce alpha particles when they react with the boron atoms in the detector gas. Either method may be used to detect neutrons in nuclear reactor. It must be noted, BF₃ counters are usually operated in the proportional region.

Fission Chamber – Wide Range Detectors

Fission chambers are ionization detectors used to detect neutrons. Fission chambers may be used as the intermediate range detectors to monitor neutron flux (reactor power) at the intermediate flux level. They also provide indication, alarms, and reactor trip signals. The design of this instrument is chosen to provide overlap between the source range channels and full span of the power range instruments.

In case of fission chambers, the chamber is coated with a thin layer of highly enriched uranium-235 to detect neutrons.  Neutrons are not directly ionizing and they have usually to be converted into charged particles before they can be detected. A thermal neutron will cause an atom of uranium-235 to fission, with the two fission fragments produced having a high kinetic energy and causing ionization of the argon gas within the detector. One advantage of using uranium-235 coating rather than boron-10 is that the fission fragments have a much higher energy than the alpha particle from a boron reaction. Therefore fission chambers are very sensitive to neutron flux and this allows the fission chambers to operate in higher gamma fields than an uncompensated ion chamber with boron lining.

Activation Foils and Flux Wires

Neutrons may be detected using activation foils and flux wires. This method is based on neutron activation, where an analyzed sample is first irradiated with neutrons to produce specific radionuclides. The radioactive decay of these produced radionuclides is specific for each element (nuclide). Each nuclide emits the characteristic gamma rays which are measured using gamma spectroscopy, where gamma rays detected at a particular energy are indicative of a specific radionuclide and determine concentrations of the elements.

Selected materials for activation foils are for example:

• indium,
• gold,
• rhodium,
• iron
• aluminum  
• niobium

These elements have large cross sections for the radiative capture of neutrons. Use of multiple absorber samples allows characterization of the neutron energy spectrum. Activation also allows recreation of an historic neutron exposure.  Commercially available criticality accident dosimeters often utilize this method. By measuring the radioactivity of thin foils, we can determine the amount of neutrons to which the foils were exposed.

Flux wires may be used in nuclear reactors to measure reactor neutron flux profiles. Principles are the same. Wire or foil is inserted directly into the reactor core, remains in the core for the length of time required for activation to the desired level. After activation, the flux wire or foil is rapidly removed from the reactor core and the activity counted. Activated foils can also discriminate energy levels by placing a cover over the foil to filter out (absorb) certain energy level neutrons. For example, cadmium is widely used to absorb thermal neutrons in a thermal neutron filters.

Detection of Fast Neutrons

Fast neutrons are neutrons of kinetic energy greater than 1 MeV (~15 000 km/s). In nuclear reactors, these neutrons are usually named fission neutrons. The fission neutrons have a Maxwell-Boltzmann distribution of energy with a mean energy (for 235U fission) 2 MeV. Inside a nuclear reactor the fast neutrons are slowed down to the thermal energies via a process called neutron moderation. These neutrons are also produced by nuclear processes such as nuclear fission or (ɑ,n) reactions.

In general, there are many detection principles and many types of detectors. Bu it must be added, detection of fast neutrons is very sophisticated discipline, since fast neutrons cross section are much smaller than in the energy range for slow neutrons. Fast neutrons are often detected by first moderating (slowing) them to thermal energies. However, during that process the information on the original energy of the neutron, its direction of travel, and the time of emission is lost.

Proton Recoil – Recoil Detectors

The most important type of detectors for fast neutrons are those which directly detect recoil particles, in particular recoil protons resulting from elastic (n, p) scattering. In fact, only hydrogen and helium nuclei are light enough for practical application.  In the latter case the recoil particles are detected in a detector. Neutrons can transfer more energy to light nuclei. This method is appropriate for detecting fast neutrons allowing detection of fast neutrons without a moderator. This methods allows the energy of the neutron to be measured together with the neutron fluence, i.e. the detector can be used as a spectrometer. Typical fast neutron detectors are liquid scintillators, helium-4 based noble gas detectors and plastic detectors (scintillators). For example, the plastic has a high hydrogen content, therefore, it is useful for fast neutron detectors, when used as a scintillator.

Bonner Spheres Spectrometer

There are several methods for detecting slow neutrons, and few methods for detecting fast neutrons. Therefore, one technique for measuring fast neutrons is to convert them to slow
neutrons, and then measure the slow neutrons. One of possible methods is based on Bonner spheres. The method was first described in 1960 by Ewing and Tom W. Bonner and employs thermal neutron detectors (usually inorganic scintillators such as ⁶LiI) embedded in moderating spheres of different sizes.  Bonner spheres have been used widely for the measurement of neutron spectra with neutron energies ranged from thermal up to at least 20 MeV. A Bonner sphere neutron spectrometer (BSS) consists of a thermal-neutron detector, a set polyethylene spherical shells and two optional lead shells of various sizes. In order to detect thermal neutrons a ³He detector or inorganic scintillators such as ⁶LiI can be used. LiGlass scintillators are very popular for detection of thermal neutrons. The advantage of LiGlass scintillators is their stability and their large range of sizes.

Detection of Neutrons using Scintillation Counter

Scintillation counters are used to measure radiation in a variety of applications including hand held radiation survey meters, personnel and environmental monitoring for radioactive contamination, medical imaging, radiometric assay, nuclear security and nuclear plant safety. They are widely used because they can be made inexpensively yet with good efficiency, and can measure both the intensity and the energy of incident radiation.

Scintillation counters can be used to detect alpha, beta, gamma radiation. They can be used also for detection of neutrons. For these purposes, different scintillators are used.

• Neutrons. Since the neutrons are electrically neutral particles, they are mainly subject to strong nuclear forces but not to electric forces. Therefore neutrons are not directly ionizing and they have usually to be converted into charged particles before they can be detected. Generally every type of neutron detector must be equipped with converter (to convert neutron radiation to common detectable radiation) and one of the conventional radiation detectors (scintillation detector, gaseous detector, semiconductor detector, etc.).  Fast neutrons (>0.5 MeV) primarily rely on the recoil proton in (n,p) reactions. Materials rich in hydrogen, for example plastic scintillators, are therefore best suited for their detection. Thermal neutrons rely on nuclear reactions such as the (n,γ) or (n,α) reactions, to produce ionization. Materials such as LiI(Eu) or glass silicates are therefore particularly well-suited for the detection of thermal neutrons. The advantage of 6LiGlass scintillators is their stability and their large range of sizes.

See also:

Application of Neutrons

See also:

Neutron

See also:

Shielding of Neutrons

• Limiting Time. The amount of radiation exposure depends directly (linearly) on the time people spend near the source of radiation. The dose can be reduced by limiting exposure time.
• Distance. The amount of radiation exposure depends on the distance from the source of radiation. Similarly to a heat from a fire, if you are too close, the intensity of heat radiation is high and you can get burned. If you are at the right distance, you can withstand there without any problems and moreover it is comfortable. If you are too far from heat source, the insufficiency of heat can also hurt you. This analogy, in a certain sense, can be applied to radiation also from nuclear sources.
• Shielding. Finally, if the source is too intensive and time or distance do not provide sufficient radiation protection the shielding must be used. Radiation shielding usually consist of barriers of lead, concrete or water. Even depleted uranium can be used as a good protection from gamma radiation, but on the other hand uranium is absolutely inappropriate shielding of neutron radiation. In short, it depends on type of radiation to be shielded, which shielding will be effective or not.

Shielding of Neutrons

There are three main features of neutrons, which are crucial in the shielding of neutrons.

• Neutrons have no net electric charge, therefore they cannot be affected or stopped by electric forces. Neutrons ionize matter only indirectly, which makes neutrons highly penetrating type of radiation.
• Neutrons scatter with heavy nuclei very elastically. Heavy nuclei very hard slow down a neutron let alone absorb a fast neutron.
• An absorption of neutron (one would say shielding) causes initiation of certain nuclear reaction (e.g. radiative capture or even fission), which is accompanied by a number of other types of radiation. In short, neutrons make matter radioactive, therefore with neutrons we have to shield also the other types of radiation.

See also: Interaction of Neutrons with Matter

Principles of Neutron Shielding

The best materials for shielding neutrons must be able to:

• Slow down neutrons (the same principle as the neutron moderation). First point can be fulfilled only by material containing light atoms (e.g. hydrogen atoms), such as water, polyethylene, and concrete. The nucleus of a hydrogen nucleus contains only a proton. Since a proton and a neutron have almost identical masses, a neutron scattering on a hydrogen nucleus can give up a great amount of its energy (even entire kinetic energy of a neutron can be transferred to a proton after one collision). This is similar to a billiard. Since a cue ball and another billiard ball have identical masses, the cue ball hitting another ball can be made to stop and the other ball will start moving with the same velocity. On the other hand, if a ping pong ball is thrown against a bowling ball (neutron vs. heavy nucleus), the ping pong ball will bounce off with very little change in velocity, only a change in direction. Therefore lead is quite ineffective for blocking neutron radiation, as neutrons are uncharged and can simply pass through dense materials.

• 

  Absorb this slow neutron. Thermal neutrons can be easily absorbed by capture in materials with high neutron capture cross sections (thousands of barns) like boron, lithium or cadmium. Generally, only a thin layer of such absorbator is sufficient to shield thermal neutrons. Hydrogen (in the form of water), which can be used to slow down neutrons, have absorbtion cross-section 0.3 barns. This is not enough, but this insufficiency can be offset by sufficient thickness of water shield.

• Shield the accompanying radiation. In the case of cadmium shield the absorption of neutrons is accompanied by strong emission of gamma rays. Therefore additional shield is necessary to attenuate the gamma rays. This phenomenon practically does not exist for lithium and is much less important for boron as a neutron absorption material. For this reason, materials containing boron are used often in neutron shields. In addition, boron (in the form of boric acid) is well soluble in water making this combination very efective neutron shield.

Water as a neutron shield

Water due to the high hydrogen content and the availability is efective and common neutron shielding. However, due to the low atomic number of hydrogen and oxygen, water is not acceptable shield against the gamma rays. On the other hand in some cases this disadvantage (low density) can be compensated by high thickness of the water shield.  In case of neutrons, water perfectly moderates neutrons, but with absorption of neutrons by hydrogen nucleus secondary gamma rays with the high energy are produced. These gamma rays highly penetrates matter and therefore it can increase requirements on the thickness of the water shield. Adding a boric acid can help with this problem (neutron absorbtion on boron nuclei without strong gamma emission), but results in another problems with corrosion of construction materials.

Concrete as a neutron shield

Most commonly used neutron shielding in many sectors of the nuclear science and engineering is shield of concrete. Concrete is also hydrogen-containing material, but unlike water concrete have higher density (suitable for secondary gamma shielding) and does not need any maintenance. Because concrete is a mixture of several different materials its composition is not constant. So when referring to concrete as a neutron shielding material, the material used in its composition should be told correctly. Generally concrete are divided to “ordinary “ concrete and “heavy” concrete. Heavy concrete uses heavy natural aggregates such as barites  (barium sulfate) or magnetite or manufactured aggregates such as iron, steel balls, steel punch or other additives. As a result of these additives, heavy concrete have higher density than ordinary concrete (~2300 kg/m³). Very heavy concrete can achieve density up to 5,900 kg/m³ with iron additives or up to 8900 kg/m³ with lead additives. Heavy concrete provide very effective protection against neutrons.

See also:

Detection of Neutrons

See also:

Neutron

See also:

Neutron Sources

• Significance of the source
• Intensity. The rate of neutrons emitted by the source.
• Energy distribution of emitted neutrons.
• Angular distribution of emitted neutrons.
• Mode of emission. Continuous or pulsed operation.

Classification by significance of the source

• Large (Significant) neutron sources
  • Nuclear Reactors. There are nuclei that can undergo fission on their own spontaneously, but only certain nuclei, like uranium-235, uranium-233 and plutonium-239, can sustain a fission chain reaction. This is because these nuclei release neutrons when they break apart, and these neutrons can induce fission of other nuclei. Uranium-235 which exists as 0.7% of naturally occurring uranium undergoes nuclear fission with thermal neutrons with the production of, on average, 2.4 fast neutrons and the release of ~ 180 MeV of energy per fission. Free neutrons released by each fission play very important role as a trigger of the reaction, but they can be also used fo another purpose. For example: One neutron is required to trigger a further fission. Part of free neutrons (let say 0.5 neutrons/fission) is absorbed in other material, but an excess of neutrons (0.9 neutrons/fission) is able to leave the surface of the reactor core and can be used as a neutron source.

• Fusion Systems. Nuclear fusion is a nuclear reaction in which two or more atomic nuclei (e.g. D+T) collide at a very high energy and fuse together. Thy byproduct of DT fusion is a free neutron (see picture), therefore also nuclear fusion reaction has the potential to produces large quantities of neutrons.
• Spallation Sources. A spallation source is a high-flux neutron source in which protons that have been accelerated to high energies hit a heavy target material, causing the emission of neutrons. The reaction occurs above a certain energy threshold for the incident particle, which is typically 5 – 15 MeV.

• Medium neutron sources
  • Bremssstrahlung from Electron Accelerators / Photofission. Energetic electrons when slowed down rapidly in a heavy target emit intense gamma radiation during the deceleration process. This is known as Bremsstrahlung or braking radiation. The interaction of the gamma radiation with the target produces neutrons via the (γ,n) reaction, or the (γ,fission) reaction when a fissile target is used. e-→Pb → γ→ Pb →(γ,n) and (γ,fission). The Bremsstrahlung γ energy exceeds the binding energy of the “last” neutron in the target. A source strength of 10¹³ neutrons/second produced in short (i.e. < 5 μs) pulses can be readily realised.
  • Dense plasme focus. The dense plasma focus (DPF) is a device that is known as an efficient source of neutrons from fusion reactions. Mechanism of dense plasma focus (DPF) is based on nuclear fusion of short-lived plasma of deuterium and/or tritium. This device produces a short-lived plasma by electromagnetic compression and acceleration that is called a pinch. This plasma is during the pinch hot and dense enough to cause nuclear fusion and the emission of neutrons.
  • Light ion accelerators. Neutrons can be also produced by particle accelerators using targets of deuterium, tritium, lithium, beryllium, and other low-Z materials. In this case the target must be bombarded with accelerated hydrogen (H), deuterium (D), or tritium (T) nuclei.

• Small neutron sources
  • Neutron Generators. Neutrons are produced in the fusion of deuterium and tritium in the following exothermic reaction. ²D + ³T → ⁴He + n + 17.6 MeV.  The neutron is produced with a kinetic energy of 14.1 MeV. This can be achieved on a small scale in the laboratory with a modest 100 kV accelerator for deuterium atoms bombarding a tritium target. Continuous neutron sources of ~10¹¹ neutrons/second can be achieved relatively simply.
  • Radioisotope source – (α,n) reactions. In certain light isotopes the ‘last’ neutron in the nucleus is weakly bound and is released when the compound nucleus formed following α-particle bombardment decays. The bombardment of beryllium by α-particles leads to the production of neutrons by the following exothermic reaction: ⁴He + ⁹Be→¹²C + n + 5.7 MeV. This reaction yields a weak source of neutrons with an energy spectrum resembling that from a fission source and is used nowadays in portable neutron sources. Radium, plutonium or americium can be used as an α-emitter.
  • Radioisotope source – (γ,n) reactions. (γ,n) reactions can also be used for the same purpose. In this type of source, because of the greater range of the γ-ray, the two physical  components of the source can be separated making it possible to ‘switch off’ the reaction if so required by removing the radioactive source from the beryllium. (γ,n) sources produce a monoenergetic neutrons unlike (α,n) sources.  The (γ,n) source uses antimony-124 as the gamma emitter in the following endothermic reaction.

¹²⁴Sb→¹²⁴Te + β− + γ

γ + ⁹Be→⁸Be + n – 1.66 MeV

• • Radioisotope source – spontaneous fission. Certain isotopes undergo spontaneous fission with emission of neutrons. The most commonly used spontaneous fission source is the radioactive isotope californium-252. Cf-252 and all other spontaneous fission neutron sources are produced by irradiating uranium or another transuranic element in a nuclear reactor, where neutrons are absorbed in the starting material and its subsequent reaction products, transmuting the starting material into the SF isotope.

See also:

Shielding of Neutrons

See also:

Neutron

What is Nuclear Energy

Nuclear energy comes either from spontaneous nuclei conversions or induced nuclei conversions. Among these conversions (nuclear reactions) belong for example nuclear fission, nuclear decay and nuclear fusion. Conversions are associated with mass and energy changes. One of the striking results of Einstein’s theory of relativity is that mass and energy are equivalent and convertible, one into the other. Equivalence of the mass and energy is described by Einstein’s famous formula:

, where M is the small amount of mass and C is the speed of light.

What that means? If the nuclear energy is generated (splitting atoms, nuclear fussion), a small amount of mass (saved in the nuclear binding energy) transforms into the pure energy (such as kinetic energy, thermal energy, or radiant energy).

The energy equivalent of one gram (1/1000 of a kilogram) of mass is equivalent to:

• 89.9 terajoules
• 25.0 million kilowatt-hours (≈ 25 GW·h)
• 21.5 billion kilocalories (≈ 21 Tcal)
• 85.2 billion BTUs

or to the energy released by combustion of the following:

• 21.5 kilotons of TNT-equivalent energy (≈ 21 kt)
• 568,000 US gallons of automotive gasoline

Any time energy is generated, the process can be evaluated from an E = mc² perspective.

Nuclear Binding Energy – Mass Defect

Conservation of Mass-Energy

At the beginning of the 20th century, the notion of mass underwent a radical revision. Mass lost its absoluteness. One of the striking results of Einstein’s theory of relativity is that mass and energy are equivalent and convertible one into the other. Equivalence of the mass and energy is described by Einstein’s famous formula E = mc². In words, energy equals mass multiplied by the speed of light squared. Because the speed of light is a very large number, the formula implies that any small amount of matter contains a very large amount of energy. The mass of an object was seen to be equivalent to energy, to be interconvertible with energy, and to increase significantly at exceedingly high speeds near that of light. The total energy of an object was understood to comprise its rest mass as well as its increase of mass caused by increase in kinetic energy.

In special theory of relativity certain types of matter may be created or destroyed, but in all of these processes, the mass and energy associated with such matter remains unchanged in quantity. It was found the rest mass an atomic nucleus is measurably smaller than the sum of the rest masses of its constituent protons, neutrons and electrons. Mass was no longer considered unchangeable in the closed system. The difference is a measure of the nuclear binding energy which holds the nucleus together. According to the Einstein relationship (E = mc²) this binding energy is proportional to this mass difference and it is known as the mass defect.

E=mc² represents the new conservation principle – the conservation of mass-energy.

Nuclear Binding Curve

If the splitting releases energy and the fusion releases the energy, so where is the breaking point? For understanding this issue it is better to relate the binding energy to one nucleon, to obtain nuclear binding curve. The binding energy per one nucleon is not linear. There is a peak in the binding energy curve in the region of stability near iron and this means that either the breakup of heavier nuclei than iron or the combining of lighter nuclei than iron will yield energy.

The reason the trend reverses after iron peak is the growing positive charge of the nuclei. The electric force has greater range than strong nuclear force. While the strong nuclear force binds only close neighbors the electric force of each proton repels the other protons.

Example: Mass defect of a 63Cu

Calculate the mass defect of a ⁶³Cu nucleus if the actual mass of ⁶³Cu in its nuclear ground state is 62.91367 u.

⁶³Cu nucleus has 29 protons and also has (63 – 29) 34 neutrons.

The mass of a proton is 1.00728 u and a neutron is 1.00867 u.

The combined mass is: 29 protons x (1.00728 u/proton) + 34 neutrons x (1.00867 u/neutron) = 63.50590 u

The mass defect is Δm = 63.50590 u – 62.91367 u =  0.59223 u

Convert the mass defect into energy (nuclear binding energy).

(0.59223 u/nucleus) x (1.6606 x 10^-27 kg/u) = 9.8346 x 10^-28 kg/nucleus

ΔE = Δmc²

ΔE = (9.8346 x 10^-28 kg/nucleus) x (2.9979 x 10⁸ m/s)² = 8.8387 x 10^-11 J/nucleus

The energy calculated in the previous example is the nuclear binding energy.  However, the nuclear binding energy may be expressed as kJ/mol (for better understanding).

Calculate the nuclear binding energy of 1 mole of ⁶³Cu:

(8.8387 x 10^-11 J/nucleus) x (1 kJ/1000 J) x (6.022 x 10²³ nuclei/mol) = 5.3227 x 10¹⁰ kJ/mol of nuclei.

One mole of ⁶³Cu (~63 grams) is bound by the nuclear binding energy (5.3227 x 10¹⁰ kJ/mol) which is equivalent to:

• 14.8 million kilowatt-hours (≈ 15 GW·h)
• 336,100 US gallons of automotive gasoline

Example: Mass defect of the reactor core

Calculate the mass defect of the 3000MW_th reactor core after one year of operation.

It is known the average recoverable energy per fission is about 200 MeV, being the total energy minus the energy of the energy of antineutrinos that are radiated away.

The reaction rate per entire 3000MW_th reactor core is about  9.33×10¹⁹ fissions / second.

The overall energy release in the units of joules is:

200×10⁶ (eV) x 1.602×10^-19 (J/eV) x 9.33×10¹⁹ (s^-1) x 31.5×10⁶ (seconds in year) = 9.4×10¹⁶ J/year

The mass defect is calculated as:

Δm = ΔE/c²

Δm = 9.4×10¹⁶ / (2.9979 x 10⁸)² = 1.046 kg

That means in a typical 3000MWth reactor core about 1 kilogram of matter is converted into pure energy.

Note that, a typical annual uranium load for a 3000MWth reactor core is about 20 tonnes of enriched uranium (i.e. about 22.7 tonnes of UO₂). Entire reactor core may contain about 80 tonnes of enriched uranium.

Mass defect directly from E=mc²

The mass defect can be calculated directly from the Einstein relationship (E = mc²) as:

Δm = ΔE/c²

Δm = 3000×10⁶ (W = J/s) x 31.5×10⁶ (seconds in year) / (2.9979 x 10⁸)² = 1.051 kg

Nuclear Energy and Electricity Production

Today we use the nuclear energy to generate useful heat and electricity. This electricity is generated in nuclear power plants. The heat source in the nuclear power plant is a nuclear reactor. As is typical in all conventional thermal power stations the heat is used to generate steam which drives a steam turbine connected to a generator which produces electricity. In 2011 nuclear power provided 10% of the world’s electricity. In 2007, the IAEA reported there were 439 nuclear power reactors in operation in the world, operating in 31 countries. They produce base-load electricity 24/7 without emitting any pollutants into the atmosphere (this includes CO2).

Nuclear Energy Consumption – Summary

Consumption of a 3000MWth (~1000MWe) reactor (12-months fuel cycle)

It is an illustrative example, following data do not correspond to any reactor design.

• Typical reactor may contain about 165 tonnes of fuel (including structural material)
• Typical reactor may contain about 100 tonnes of enriched uranium (i.e. about 113 tonnes of uranium dioxide).
• This fuel is loaded within, for example, 157 fuel assemblies composed of over 45,000 fuel rods.
• A common fuel assembly contain energy for approximately 4 years of operation at full power.
• Therefore about one quarter of the core is yearly removed to spent fuel pool (i.e. about 40 fuel assemblies), while the remainder is rearranged to a location in the core better suited to its remaining level of enrichment (see Power Distribution).
• The removed fuel (spent nuclear fuel) still contains about 96% of reusable material (it must be removed due to decreasing k_inf of an assembly).

• Annual natural uranium consumption of this reactor is about 250 tonnes of natural uranium (to produce of about 25 tonnes of enriched uranium).

• Annual enriched uranium consumption of this reactor is about 25 tonnes of enriched uranium.

• Annual fissile material consumption of this reactor is about 1 005 kg.

• Annual matter consumption of this reactor is about 1.051 kg.

• But it corresponds to about 3 200 000 tons of coal burned in coal-fired power plant per year.

See also: Fuel Consumption

See also:

Atomic and Nuclear Structure

See also:

Atomic and Nuclear Physics

See also:

Radiation

Basics of Nuclear Fission

Basics of Nuclear Fission

There are nuclei that can undergo fission on their own spontaneously, but only certain nuclei, like uranium-235, uranium-233 and plutonium-239, can sustain a fission chain reaction. This is because these nuclei release neutrons when they break apart, and these neutrons can induce fission of other nuclei. Free neutrons released by each fission play very important role as a trigger of the reaction.

Chain Reaction

Chain reaction

A nuclear chain reaction occurs when one single nuclear reaction causes an average of one or more subsequent nuclear reactions, thus leading to the possibility of a self-propagating series of these reactions. The “one or more” is the key parameter of reactor physics. To raise or lower the power, the amount of reactions must be changed (using the control rods) so that the number of neutrons present (and hence the rate of power generation) is either reduced or increased.

Youtube animation

Principles of Nuclear Fission

In general, the neutron-induced fission reaction is the reaction, in which the incident neutron enters the heavy target nucleus (fissionable nucleus), forming a compound nucleus that is excited to such a high energy level (E_excitation > E_critical) that the nucleus splits into two large fission fragments. A large amount of energy is released in the form of radiation and fragment kinetic energy. Moreover and what is crucial, the fission process may produce 2, 3 or more free neutrons and these neutrons can trigger further fission and a chain reaction can take place. In order to understand the process of fission, we must understand processes, that occur inside the nucleus to be fissioned. At first, the nuclear binding energy must be defined.

Nuclear Binding Energy

During the nuclear splitting or nuclear fusion, some of the mass of the nucleus gets converted into huge amounts of energy and thus this mass is removed from the total mass of the original particles, and the mass is missing in the resulting nucleus. The nuclear binding energies are enormous, they are on the order of a million times greater than the electron binding energies of atoms.

For a nucleus with A (mass number) nucleons, the binding energy per nucleon E_b/A can be calculated. This calculated fraction is shown in the chart as a function of them mass number A. As can be seen, for low mass numbers E_b/A increases rapidly and reaches a maximum of 8.8 MeV at approximately A=60. The nuclei with the highest binding energies, that are most tightly bound belong to the “iron group” of isotopes (⁵⁶Fe, ⁵⁸Fe, ⁶²Ni). After that, the binding energy per nucleon decreases. In the heavy nuclei (A>60) region, a more stable configuration is obtained, when a heavy nucleus splits into two lighter nuclei. This is the origin of the fission process. It may seem that all the heavy nuclei may undergo fission or even spontaneous fission. In fact, for all nuclei with atomic number greater than about 60, fission occurs very rarely. In order to fission process to take place, a sufficient amount of energy must be added to the nucleus and no matter how. The energetics and binding energies of certain nucleus are well described by the Liquid Drop Model, which examines the global properties of nuclei.

Liquid Drop Model

One of the first models which could describe very well the behavior of the nuclear binding energies and therefore of nuclear masses was the mass formula of von Weizsaecker (also called the semi-empirical mass formula – SEMF), that was published in 1935 by German physicist Carl Friedrich von Weizsäcker. This theory is based on the liquid drop model proposed by George Gamow.

According to this model, the atomic nucleus behaves like the molecules in a drop of liquid. But in this nuclear scale, the fluid is made of nucleons (protons and neutrons), which are held together by the strong nuclear force. The liquid drop model of the nucleus takes into account the fact that the nuclear forces on the nucleons on the surface are different from those on nucleons in the interior of the nucleus. The interior nucleons are completely surrounded by other attracting nucleons. Here is the analogy with the forces that form a drop of liquid.

In the ground state the nucleus is spherical. If the sufficient kinetic or binding energy is added, this spherical nucleus may be distorted into a dumbbell shape and then may be splitted into two fragments. Since these fragments are a more stable configuration, the splitting of such heavy nuclei must be accompanied by energy release. This model does not explain all the properties of the atomic nucleus, but does explain the predicted nuclear binding energies.

The nuclear binding energy as a function of the mass number A and the number of
protons Z based on the liquid drop model can be written as:

This formula is called the Weizsaecker Formula (or the semi-empirical mass formula). The physical meaning of this equation can be discussed term by term.

Volume term

Volume term – a_V.A. The first two terms describe a spherical liquid drop of an incompressible fluid with a contribution from the volume scaling with A and from the surface, scaling with A^2/3. The first positive term a_V.A is known as the volume term and it is caused by the attracting strong forces between the nucleons. The strong force has a very limited range and a given nucleon may only interact with its direct neighbours. Therefore this term is proportional to A, instead of A². The coefficient a_V is usually about ~ 16 MeV.

Surface term

Surface term – a_sf.A^2/3. The surface term is also based on the strong force, it is, in fact, a correction to the volume term. The point is that particles at the surface of the nucleus are not completely surrounded by other particles. In the volume term, it is suggested that each nucleon interacts with a constant number of nucleons, independent of A. This assumption is very nearly true for nucleons deep within the nucleus, but causes an overestimation of the binding energy on the surface. By analogy with a liquid drop this effect is indicated as the surface tension effect. If the volume of the nucleus is proportional to A, then the geometrical radius should be proportional to A^1/3 and therefore the surface term must be proportional to the surface area i.e. proportional to A^2/3.

Coulomb term

Coulomb term – a_C.Z².A^-⅓. This term describes the Coulomb repulsion between the uniformly distributed protons and is proportional to the number of proton pairs Z²/R, whereby R is proportional to A^1/3. This effect lowers the binding energy because of the repulsion between charges of equal sign.

Asymmetry term

Asymmetry term – a_A.(A-2Z)²/A. This term cannot be described as ‘classically’ as the first three. This effect is not based on any of the fundamental forces, this effect is based only on the Pauli exclusion principle (no two fermions can occupy exactly the same quantum state in an atom). The heavier nuclei contain more neutrons than protons. These extra neutrons are necessary for stability of the heavier nuclei. They provide (via the attractive forces between the neutrons and protons) some compensation for the repulsion between the protons. On the other hand, if there are significantly more neutrons than protons in a nucleus, some of the neutrons will be higher in energy level in the nucleus. This is the basis for a correction factor, the so-called symmetry term.

Pairing term

Pairing term – δ(A,Z). The last term is the pairing term δ(A,Z). This term captures the effect of spin-coupling. Nuclei with an even number of protons and an even number of neutrons are (due to Pauli exclusion principle) very stable thanks to the occurrence of ‘paired spin’. On the other hand, nuclei with an odd number of protons and neutrons are mostly unstable.

Table of Calculated Binding Energies

In order to calculate the binding energy, the coefficients a_V, a_S, a_C, a_A and a_P must be known. The coefficients have units of megaelectronvolts (MeV) and are calculated by fitting to experimentally measured masses of nuclei. They usually vary depending on the fitting methodology. According to ROHLF, J. W., Modern Physics from α to Z0 , Wiley, 1994., the coefficients in the equation are following:

Using the Weizsaecker formula, also the mass of an atomic nucleus can be derived and is given by:

m = Z.m_p +N.m_n -E_b/c²

where m_p and m_n are the rest mass of a proton and a neutron, respectively, and E_b is the nuclear binding energy of the nucleus.

From the nuclear binding energy curve and from the table it can be seen that, in the case of splitting a ²³⁵U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is larger than that of the original ²³⁵U nucleus.
According to the Weizsaecker formula, the total energy released for such reaction will be approximately 235 x (8.5 – 7.6) ≈ 200 MeV.

See also: Liquid Drop Model

Critical Energy – Threshold Energy for Fission

In principle, any nucleus, if brought into sufficiently high excited state, can be splitted. For fission to occur, the excitation energy must be above a particular value for certain nuclide. The minimum excitation energy required for fission to occur is known as the critical energy (E_crit) or threshold energy.

The critical energy depends on the nuclear structure and is quite large for light nuclei with Z < 90. For heavier nuclei with Z > 90, the critical energy is about 4 to 6 MeV for A-even nuclei, and generally is much lower for A-odd nuclei. It must be noted, some heavy nuclei (eg. ²⁴⁰Pu or ²⁵²Cf) exhibit fission even in the ground state (without externally added excitation energy). This phenomena is known as the spontaneous fission. This process occur without the addition of the critical energy by the quantum-mechanical process of quantum tunneling through the Coulomb barrier (similarly like alpha particles in the alpha decay). The spontaneous fission contributes to ensure sufficient neutron flux on source range detectors when reactor is subcritical in long term shutdown.

See also: Critical Energy – Threshold Energy for Fission

Energy Release from Fission

In general, the nuclear fission results in the release of enormous quantities of energy. The amount of energy depends strongly on the nucleus to be fissioned and also depends strongly on the kinetic energy of an incident neutron. In order to calculate the power of a reactor, it is necessary to be able precisely identify the individual components of this energy. At first, it is important to distinguish between the total energy released and the energy that can be recovered in a reactor.

The total energy released in fission can be calculated from binding energies of initial target nucleus to be fissioned and binding energies of fission products. But not all the total energy can be recovered in a reactor. For example, about 10 MeV is released in the form of neutrinos (in fact antineutrinos). Since the neutrinos are weakly interacting (with extremely low cross-section of any interaction), they do not contribute to the energy that can be recovered in a reactor.

In order to understand this issue, we have to first investigate a typical fission reaction such as the one listed below.

Using this picture, we can identify and also describe almost all the individual components of the total energy energy released during the fission reaction.

As can be seen from the description of the individual components of the total energy energy released during the fission reaction, there is significant amount of energy generated outside the nuclear fuel (outside fuel rods). Especially the kinetic energy of prompt neutrons is largely generated in the coolant (moderator). This phenomena needs to be included in the nuclear calculations.

For LWR, it is generally accepted that about 2.5% of total energy is recovered in the moderator. This fraction of energy depends on the materials, their arrangement within the reactor, and thus on the reactor type.

Fission Fragments – Products of Nuclear Fission

Nuclear fission fragments are the fragments left after a nucleus fissions. Typically, when uranium 235 nucleus undergoes fission, the nucleus splits into two smaller nuclei, along with a few neutrons and release of energy in the form of heat (kinetic energy of the these fission fragments) and gamma rays. The average of the fragment mass is about 118, but very few fragments near that average are found. It is much more probable to break up into unequal fragments, and the most probable fragment masses are around mass 95 (Krypton) and 137 (Barium).

Most of these fission fragments are highly unstable (radioactive) and undergo further radioactive decays to stabilize itself. Fission fragments interact strongly with the surrounding atoms or molecules traveling at high speed, causing them to ionize.

See also: Interaction of Heavy Charged Particles with Matter

Prompt and Delayed Neutrons

It is known the fission neutrons are of importance in any chain-reacting system. Neutrons trigger the nuclear fission of some nuclei (²³⁵U, ²³⁸U or even ²³²Th). What is crucial the fission of such nuclei produces 2, 3 or more free neutrons.

But not all neutrons are released at the same time following fission. Even the nature of creation of these neutrons is different. From this point of view we usually divide the fission neutrons into two following groups:

• Prompt Neutrons. Prompt neutrons are emitted directly from fission and they are emitted within very short time of about 10^-14 second.
• Delayed Neutrons. Delayed neutrons are emitted by neutron rich fission fragments that are called the delayed neutron precursors. These precursors usually undergo beta decay but a small fraction of them are excited enough to undergo neutron emission. The fact the neutron is produced via this type of decay and this happens orders of magnitude later compared to the emission of the prompt neutrons, plays an extremely important role in the control of the reactor.

See also: Prompt Neutrons

See also: Delayed Neutrons

See also: Reactor control with and without delayed neutrons – Interactive chart

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

Key Characteristics of Prompt Neutrons

• Prompt neutrons are emitted directly from fission and they are emitted within very short time of about 10^-14 second.

• Most of the neutrons produced in fission are prompt neutrons – about 99.9%.

• For example a fission of ²³⁵U by thermal neutron yields 2.43 neutrons, of which 2.42 neutrons are prompt neutrons and 0.01585 neutrons are the delayed neutrons.

• The production of prompt neutrons slightly increase with incident neutron energy.

• Almost all prompt fission neutrons have energies between 0.1 MeV and 10 MeV.

• The mean neutron energy is about 2 MeV. The most probable neutron energy is about 0.7 MeV.

• In reactor design the prompt neutron lifetime (PNL) belongs to key neutron-physical characteristics of reactor core.

• Its value depends especially on the type of the moderator and on the energy of the neutrons causing fission.

• In an infinite reactor (without escape) prompt neutron lifetime is the sum of the slowing down time and the diffusion time.

• In LWRs the PNL increases with the fuel burnup.

• The typical prompt neutron lifetime in thermal reactors is on the order of 10^-4 second.

• The typical prompt neutron lifetime in fast reactors is on the order of 10^-7 second.

Key Characteristics of Delayed Neutrons

• The presence of delayed neutrons is perhaps most important aspect of the fission process from the viewpoint of reactor control.

• Delayed neutrons are emitted by neutron rich fission fragments that are called the delayed neutron precursors.

• These precursors usually undergo beta decay but a small fraction of them are excited enough to undergo neutron emission.

• The emission of neutron happens orders of magnitude later compared to the emission of the prompt neutrons.

• About 240 n-emitters are known between ⁸He and ²¹⁰Tl, about 75 of them are in the non-fission region.

• In order to simplify reactor kinetic calculations it is suggested to group together the precursors based on their half-lives.

• Therefore delayed neutrons are traditionally represented by six delayed neutron groups.

• Neutrons can be produced also in (γ, n) reactions (especially in reactors with heavy water moderator) and therefore they are usually referred to as photoneutrons. Photoneutrons are usually treated no differently than regular delayed neutrons in the kinetic calculations.

• The total yield of delayed neutrons per fission, v_d, depends on:
  • Isotope, that is fissioned.
  • Energy of a neutron that induces fission.

• Variation among individual group yields is much greater than variation among group periods.

• In reactor kinetic calculations it is convenient to use relative units usually referred to as delayed neutron fraction (DNF).

• At the steady state condition of criticality, with k_eff = 1, the delayed neutron fraction is equal to the precursor yield fraction β.

• In LWRs the β decreases with fuel burnup. This is due to isotopic changes in the fuel.

• Delayed neutrons have initial energy between 0.3 and 0.9 MeV with an average energy of 0.4 MeV.

• Depending on the type of the reactor, and their spectrum, the delayed neutrons may be more (in thermal reactors) or less effective than prompt neutrons (in fast reactors). In order to include this effect into the reactor kinetic calculations the effective delayed neutron fraction – β_eff must be defined.

• The effective delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor β_eff = β . I.

• The weighted delayed generation time is given by τ = ∑_iτ_i . β_i / β = 13.05 s, therefore the weighted decay constant λ = 1 / τ ≈ 0.08 s^-1.

• The mean generation time with delayed neutrons is about ~0.1 s, rather than ~10^-5 as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

• Their presence completely changes the dynamic time response of a reactor to some reactivity change, making it controllable by control systems such as the control rods.

Capture-to-Fission Ratio

The probability that a neutron that is absorbed in a fissile nuclide causes a
fission is very important parameter of each fissile isotope. In terms of cross-sections, this probability is defined as:

σ_f / (σ_f + σ_γ) = 1 / (1 + σ_γ/σ_f) = 1 / (1 + α),

where α = σ_γ/σ_f is referred to as the capture-to-fission ratio. The capture-to-fission ratio may be used as an indicator of “quality” of fissile isotopes. The lower C/F ratio simply means that an absorption reaction will result in the fission rather than in the radiative capture. The ratio depends strongly on the incident neutron energy. In the fast neutron region, C/F ratio decreases. It is determined by the steeper decrease in radiative capture cross-section (see chart).

For ²³⁵U and ²³³U the thermal neutron capture-to-fission ratios are typically lower than those for fast neutrons (for mean energy of about 100 keV). It must be noted, the neutron flux of most fast reactors tends to peak around 200 keV, but the mean energy is between 100-200 keV depending on certain reactor design.

Further increase in neutron energy causes conversely a decrease in C/F ratio. This is not the case of ²³⁹Pu, for 100 keV neutrons, the C/F ratio is lower than for thermal neutrons. For the fissile isotopes (²³³U, ²³⁵U and ²³⁹Pu), a small capture-to-fission ratio is an advantage, because neutrons captured onto them are lost.

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

Nuclear Fission Chain Reaction

The chain reaction can take place only in the proper multiplication environment and only under proper conditions. It is obvious, if one neutron causes two further fissions, the number of neutrons in the multiplication system will increase in time and the reactor power (reaction rate) will also increase in time. In order to stabilize such multiplication environment, it is necessary to increase the non-fission neutron absorption in the system (e.g. to insert control rods). Moreover, this multiplication environment (nuclear reactor) behaves like the exponential system, that means the power increase is not linear, but it is exponential.

On the other hand, if one neutron causes less than one further fission, the number of neutrons in the multiplication system will decrease in time and the reactor power (reaction rate) will also decrease in time. In order to sustain the chain reaction, it is necessary to decrease the non-fission neutron absorption in the system (e.g. to withdraw control rods).

In fact, there is always a competition for the fission neutrons in the multiplication environment, some neutrons will cause further fission reaction, some will be captured by fuel materials or non-fuel materials and some will leak out of the system.

In order to describe the multiplication system, it is necessary to define the infinite and finite multiplication factor of a reactor. The method of calculations of multiplication factors has been developed in the early years of nuclear energy and is only applicable to thermal reactors, where the bulk of fission reactions occurs at thermal energies. This method well puts into the context all the processes, that are associated with the thermal reactors (e.g. the neutron thermalisation, the neutron diffusion or the fast fission), because the most important neutron-physical processes occur in energy regions that can be clearly separated from each other. In short, the calculation of multiplication factor gives a good insight in the processes that occur in each thermal multiplying system.

Distinction between Fissionable, Fissile and Fertile

In nuclear engineering, fissionable material (nuclide) is material  that is capable of undergoing fission reaction after absorbing either thermal (slow or low energy) neutron or fast (high energy) neutron. Fissionable materials are a superset of fissile materials. Fissionable materials also include an isotope ²³⁸U that can be fissioned only with high energy (>1MeV) neutron. These materials are used to fuel thermal nuclear reactors, because they are capable of sustaining a nuclear fission chain reaction.

Fissile materials undergoes fission reaction after absorption of the binding energy of thermal neutron. They do not require additional kinetic energy for fission. If the neutron has higher kinetic energy, this energy will be transformed into additional excitation energy of the compound nucleus. On the other hand, the binding energy released by compound nucleus of (²³⁸U + n) after absorption of thermal neutron is less than the critical energy, so the fission reaction cannot occur. The distinction is described in the following points.

• Fissile materials are a subset of fissionable materials.
• Fissionable material consist of isotopes that are capable of undergoing nuclear fission after capturing either fast neutron (high energy neutron – let say >1 MeV) or thermal neutron (low energy neutron – let say 0.025 eV).   Typical fissionable materials: ²³⁸U, ²⁴⁰Pu, but also ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu
• Fissile material consist of fissionable isotopes that are capable of undergoing nuclear fission only after capturing a thermal neutron. ²³⁸U is not fissile isotope, because ²³⁸U cannot be fissioned by thermal neutron. ²³⁸U does not meet also alternative requirement to fissile materials. ²³⁸U is not capable of sustaining a nuclear fission chain reaction, because neutrons produced by fission of ²³⁸U have lower energies than original neutron (usually below the threshold energy of 1 MeV). Typical fissile materials: ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu.
• Fertile material consist of isotopes that are not fissionable by thermal neutrons, but can be converted into fissile isotopes (after neutron absorption and subsequent nuclear decay). Typical fertile materials: ²³⁸U, ²³²Th.

Fissile materials undergoes fission reaction after absorption of the binding energy of thermal neutron. They do not require additional kinetic energy for fission. If the neutron has higher kinetic energy, this energy will be transformed into additional excitation energy of the compound nucleus. On the other hand, the binding energy released by compound nucleus of (²³⁸U + n) after absorption of thermal neutron is less than the critical energy, so the fission reaction cannot occur. The distinction is described in the following points.

• Fissile materials are a subset of fissionable materials.
• Fissionable material consist of isotopes that are capable of undergoing nuclear fission after capturing either fast neutron (high energy neutron – let say >1 MeV) or thermal neutron (low energy neutron – let say 0.025 eV).   Typical fissionable materials: ²³⁸U, ²⁴⁰Pu, but also ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu
• Fissile material consist of fissionable isotopes that are capable of undergoing nuclear fission only after capturing a thermal neutron. ²³⁸U is not fissile isotope, because ²³⁸U cannot be fissioned by thermal neutron. ²³⁸U does not meet also alternative requirement to fissile materials. ²³⁸U is not capable of sustaining a nuclear fission chain reaction, because neutrons produced by fission of ²³⁸U have lower energies than original neutron (usually below the threshold energy of 1 MeV). Typical fissile materials: ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu.
• Fertile material consist of isotopes that are not fissionable by thermal neutrons, but can be converted into fissile isotopes (after neutron absorption and subsequent nuclear decay). Typical fertile materials: ²³⁸U, ²³²Th.

See also: Neutron cross-section

Comparison of cross-sections

Source: JANIS (Java-based nuclear information software)  http://www.oecd-nea.org/janis/

Fissile / Fertile Material Cross-sections. Comparison of total fission cross-sections.

Uranium 238. Comparison of total fission cross-section and cross-section for radiative capture.

See also:

See also:

Neutron Nuclear Reactions

See also:

A nuclear reaction is considered to be the process in which two nuclear particles (two nuclei or a nucleus and a nucleon) interact to produce two or more nuclear particles or ˠ-rays (gamma rays). Thus, a nuclear reaction must cause a transformation of at least one nuclide to another. Sometimes if a nucleus interacts with another nucleus or particle without changing the nature of any nuclide, the process is referred to a nuclear scattering, rather than a nuclear reaction. Perhaps the most notable nuclear reactions are the nuclear fusion reactions of light elements that power the energy production of stars and the Sun. Natural nuclear reactions occur also in the interaction between cosmic rays and matter.

The most notable man-controlled nuclear reaction is the fission reaction which occurs in nuclear reactors. Nuclear reactors are devices to initiate and control a nuclear chain reaction, but there are not only manmade devices. The world’s first nuclear reactor operated about two billion years ago. The natural nuclear reactor formed at Oklo in Gabon, Africa, when a uranium-rich mineral deposit became flooded with groundwater that acted as a neutron moderator, and a nuclear chain reaction started.  These fission reactions were sustained for hundreds of thousands of years, until a chain reaction could no longer be supported. This was confirmed by existence of isotopes of the fission-product gas xenon and by different ratio of U-235/U-238 (enrichment of natural uranium).

See also: TALYS – a software for the simulation of nuclear reactions.

See also: JANIS – Java-based Nuclear Data Information System

Notation of Nuclear Reactions

Standard nuclear notation shows (see picture) the chemical symbol, the mass number and the atomic number of the isotope.

If the initial nuclei are denoted by a and b, and the product nuclei are denoted by c and d, the reaction can be represented by the equation:

 a + b → c + d

Instead of using the full equations in the style above, in many situations a compact notation is used to describe nuclear reactions. This style of the form a(b,c)d is equivalent to a + b producing c + d. Light particles are often abbreviated in this shorthand, typically p means proton, n means neutron, d means deuteron, α means an alpha particle or helium-4, β means beta particle or electron, γ means gamma photon, etc. The reaction above would be written as 10B(n,α)7Li.

Basic Classification of Nuclear Reactions

In order to understand the nature of neutron nuclear reactions, the classification according to the time scale of of these reactions has to be introduced. Interaction time is critical for defining the reaction mechanism.

There are two extreme scenarios for nuclear reactions (not only neutron reactions):

• A projectile and a target nucleus are within the range of nuclear forces for the very short time allowing for an interaction of a single nucleon only. These type of reactions are called the direct reactions.
• A projectile and a target nucleus are within the range of nuclear forces for the time allowing for a large number of interactions between nucleons. These type of reactions are called the compound nucleus reactions.

In fact, there is always some non-direct (multiple internuclear interaction) component in all reactions, but the direct reactions have this component limited.

Types of Nuclear Reactions

Although the number of possible nuclear reactions is enormous, nuclear reactions can be sorted by types. Most of nuclear reactions are accompanied by gamma emission. Some examples are:

• Elastic scattering. Occurs, when no energy is transferred between the target nucleus and the incident particle.

 208Pb (n, n) 208Pb

•  Inelastic scattering. Occurs, when energy is transferred. The difference of kinetic energies is saved in excited nuclide.

 40Ca (α, α’) 40mCa

• Capture reactions. Both charged and neutral particles can be captured by nuclei. This is accompanied by the emission of ˠ-rays. Neutron capture reaction produces radioactive nuclides (induced radioactivity).

 238U (n, ˠ) 239U

• Transfer Reactions. The absorption of a particle accompanied by the emission of one or more particles is called the transfer reaction.

4He (α, p) 7Li

• Fission reactions. Nuclear fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts (lighter nuclei). The fission process often produces free neutrons and photons (in the form of gamma rays), and releases a large amount of energy.

235U (n, 3 n) fission products

• Fusion reactions.  Occur when, two or more atomic nuclei collide at a very high speed and join to form a new type of atomic nucleus.The fusion reaction of deuterium and tritium is particularly interesting because of its potential of providing energy for the future.

3T (d, n) 4He

• Spallation reactions. Occur, when a nucleus is hit by a particle with sufficient energy and momentum to knock out several small fragments or, smash it into many fragments.

• Nuclear decay (Radioactive decay). Occurs when an unstable atom loses energy by emitting ionizing radiation. Radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. There are many types of radioactive decay:
  • Alpha radioactivity. Alha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Because of its very large mass (more than 7000 times the mass of the beta particle) and its charge, it heavy ionizes material and has a very short range.

• • Beta radioactivity. Beta particles are high-energy, high-speed electrons or positrons emitted by certain types of radioactive nuclei such as potassium-40. The beta particles have greater range of penetration than alpha particles, but still much less than gamma rays.The beta particles emitted are a form of ionizing radiation also known as beta rays. The production of beta particles is termed beta decay.

• • Gamma radioactivity. Gamma rays are electromagnetic radiation of an very high frequency and are therefore high energy photons. They are produced by the decay of nuclei as they transition from a high energy state to a lower state known as gamma decay. Most of nuclear reactions are accompanied by gamma emission.

• • Neutron emission. Neutron emission is a type of radioactive decay of nuclei containing excess neutrons (especially fission products), in which a neutron is simply ejected from the nucleus. This type of radiation plays key role in nuclear reactor control, because these neutrons are delayed  neutrons.

Conservation Laws in Nuclear Reactions

In analyzing nuclear reactions, we apply the many conservation laws. Nuclear reactions are subject to classical conservation laws for charge, momentum, angular momentum, and energy(including rest energies).  Additional conservation laws, not anticipated by classical physics, are:

• Law of Conservation of Lepton Number
• Law of Conservation of Baryon Number
• Law of Conservation of Electric Charge

Law of Conservation of Lepton Number

In particle physics, the lepton number is used to denote which particles are leptons and which particles are not. Each lepton has a lepton number of 1 and each antilepton has a lepton number of -1. Other non-leptonic particles have a lepton number of 0. The lepton number is a conserved quantum number in all particle reactions. A slight asymmetry in the laws of physics allowed leptons to be created in the Big Bang.

The conservation of lepton number means that whenever a lepton of a certain generation is created or destroyed in a reaction, a corresponding antilepton from the same generation must be created or destroyed. It must be added, there is a separate requirement for each of the three generations of leptons, the electron, muon and tau and their associated neutrinos.

Consider the decay of the neutron. The reaction involves only first generation leptons: electrons and neutrinos:

Since the lepton number must be equal to zero on both sides and it was found that the reaction is a three-particle decay (the electrons emitted in beta decay have a continuous rather than a discrete spectrum),  the third particle must be an electron antineutrino.

Law of Conservation of Baryon Number

In particle physics, the baryon number is used to denote which particles are baryons and which particles are not. Each baryon has a baryon number of 1 and each antibaryonhas a baryon number of -1. Other non-baryonic particles have a baryon number of 0. Since there are exotic hadrons like pentaquarks and tetraquarks, there is a general definition of baryon number as:

where n_q is the number of quarks, and n_q is the number of antiquarks.

The baryon number is a conserved quantum number in all particle reactions.

The law of conservation of baryon number states that:

The sum of the baryon number of all incoming particles is the same as the sum of the baryon numbers of all particles resulting from the reaction.

For example, the following reaction has never been observed:

even if the incoming proton has sufficient energy and charge, energy, and so on, are conserved. This reaction does not conserve baryon number since the left side has B =+2, and the right has B =+1.

On the other hand, the following reaction (proton-antiproton pair production) does conserve B and does occur if the incoming proton has sufficient energy (the threshold energy = 5.6 GeV):

As indicated, B = +2 on both sides of this equation.

From these and other reactions, the conservation of baryon number has been established as a basic principle of physics.

This principle provides basis for the stability of the proton. Since the proton is the lightest particle among all baryons, the hypothetical products of its decay would have to be non-baryons. Thus, the decay would violate the conservation of baryon number. It must be added some theories have suggested that protons are in fact unstable with very long half-life (~10³⁰ years) and that they decay into leptons. There is currently no experimental evidence that proton decay occurs.

Law of Conservation of Electric Charge

The law of conservation of electric charge can be demonstrated also on positron-electron pair production. Since a gamma ray is electrically neutral and sum of the electric charges of electron and positron is also zero, the electric charge in this reaction is also conserved.

Ɣ → e^– + e⁺

It must be added, in order for electron-positron pair production to occur, the electromagnetic energy of the photon must be above a threshold energy, which is equivalent to the rest mass of two electrons. The threshold energy (the total rest mass of produced particles) for electron-positron pair production is equal to 1.02MeV (2 x 0.511MeV) because the rest mass of a single electron is equivalent to 0.511MeV of energy. If the original photon’s energy is greater than 1.02MeV, any energy above 1.02MeV is according to the conservation law split between the kinetic energy of motion of the two particles. The presence of an electric field of a heavy atom such as lead or uranium is essential in order to satisfy conservation of momentum and energy. In order to satisfy both conservation of momentum and energy, the atomic nucleus must receive some momentum. Therefore a photon pair production in free space cannot occur.

Some conservation principles have arisen from theoretical considerations, others are just empirical relationships. Notwithstanding, any reaction not expressly forbidden by the conservation laws will generally occur, if perhaps at a slow rate. This expectation is based on quantum mechanics. Unless the barrier between the initial and final states is infinitely high, there is always a non-zero probability that a system will make the transition between them.

For purposes of analyzing non-relativistic reactions, it is sufficient to note four of the fundamental laws governing these reactions.

1. Conservation of nucleons. The total number of nucleons before and after a reaction are the same.
2. Conservation of charge. The sum of the charges on all the particles before and after a reaction are the same
3. Conservation of momentum. The total momentum of the interacting particles before and after a reaction are the same.
4. Conservation of energy. Energy, including rest mass energy, is conserved in nuclear reactions.

Reference: Lamarsh, John R. Introduction to Nuclear engineering 2nd Edition.

Energetics of Nuclear Reactions – Q-value

In nuclear and particle physics the energetics of nuclear reactions is determined by the Q-value of that reaction. The Q-value of the reaction is defined as the difference between the sum of the masses of the initial reactants and the sum of the masses of the final products, in energy units (usually in MeV).

Consider a typical reaction, in which the projectile a and the target A gives place to two products, B and b. This can also be expressed in the notation that we used so far, a + A → B + b, or even in a more compact notation, A(a,b)B.

See also: E=mc²

The Q-value of this reaction is given by:

Q = [m_a + m_A – (m_b + m_B)]c²

which is the same as the excess kinetic energy of the final products:

Q = T_final – T_initial

   = T_b + T_B – (T_a + T_A)

For reactions in which there is an increase in the kinetic energy of the products Q is positive. The positive Q reactions are said to be exothermic (or exergic). There is a net release of energy, since the kinetic energy of the final state is greater than the kinetic energy of the initial state.

For reactions in which there is a decrease in the kinetic energy of the products Q is negative. The negative Q reactions are said to be endothermic (or endoergic) and they require a net energy input.

The energy released in a nuclear reaction can appear mainly in one of three ways:

• Kinetic energy of the products
• Emission of gamma rays. Gamma rays are emitted by unstable nuclei in their transition from a high energy state to a lower state known as gamma decay.
• Metastable state. Some energy may remain in the nucleus, as a metastable energy level.

A small amount of energy may also emerge in the form of X-rays. Generally, products of nuclear reactions may have different atomic numbers, and thus the configuration of their electron shells is different in comparison with reactants. As the electrons rearrange themselves and drop to lower energy levels, internal transition X-rays (X-rays with precisely defined emission lines) may be emitted.

See also: Q-value Calculator

Exothermic Reactions

Endothermic Reactions

Example: Endothermic Reaction - Photoneutrons

In nuclear reactors the gamma radiation plays a significant role also in reactor kinetics and in a subcriticality control. Especially in nuclear reactors with D₂O moderator (CANDU reactors) or with Be reflectors (some experimental reactors). Neutrons can be produced also in (γ, n) reactions and therefore they are usually referred to as photoneutrons.

A high energy photon (gamma ray) can under certain conditions eject a neutron from a nucleus. It occurs when its energy exceeds the binding energy of the neutron in the nucleus. Most nuclei have binding energies in excess of 6 MeV, which is above the energy of most gamma rays from fission. On the other hand there are few nuclei with sufficiently low binding energy to be of practical interest. These are: ²D, ⁹Be, ⁶Li, ⁷Li and ¹³C. As can be seen from the table the lowest threshold have ⁹Be with 1.666 MeV and ²D with 2.226 MeV.

In case of deuterium, neutrons can be produced by the interaction of gamma rays (with a minimum energy of 2.22 MeV) with deuterium:

The reaction Q-value is calculated below:

The atom masses of the reactant and products are:

m(²D) = 2.01363 amu

m(¹n) = 1.00866 amu

m(¹H) = 1.00728 amu

Using the mass-energy equivalence, we get the Q-value of this reaction as:

Q = {2.01363 [amu] – (1.00866+1.00728) [amu]} x 931.481 [MeV/amu]

= -0.00231 x 931.481 = -2.15 MeV

Example: Endothermic Reaction - (α,n) reaction

Calculate the reaction Q-value of the following reaction:

7Li (α, n) 10B

The atom masses of the reactants and products are:

m(⁴He) = 4.0026 amu

m(⁷Li) = 7.0160 amu

m(¹n) = 1.0087 amu

m(¹⁰B) = 10.01294 amu

Using the mass-energy equivalence, we get the Q-value of this reaction as:

Q = {(7.0160+4.0026) [amu] – (1.0087+10.01294) [amu]} x 931.481 [MeV/amu]

= 0.00304 x 931.481 = -2.83 MeV

Test your Knowledge – Nuclear Reactions

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See also:

Radioactive Decay

See also:

Atomic and Nuclear Physics

See also:

Binding Energy