26. Integrated cross section

Integrated cross section – is the integral from differential cross section by the angles from 0 to 2∏

The integrated cross section of nuclear reaction a + A → b + B is called a value:

where n – amount of target particle by unit of square, N₀ – amount of collided particles a, dN_b – amont of particle b, product of reaction. The integrated cross section _ab of reaction a + A → b + B and back _ba of reaction b + B → a + A connected with each other by principle of detailed balance.

where j_a, j_A, j_b, j_B are spins _, and _a и_b are momentums of particles in center mass of system. 

35. Direct Nuclear Reaction

Direct Nuclear Reaction  a nuclear process in which energy introduced into the atomic nucleus is imparted preferentially to one nucleon or to a small group of nucleons. A number of types exist. Direct nuclear reactions can be caused by all possible incident particles, from gamma quanta to multiply charged ions, over a wide range of energies (from a few million to several billion electron volts). They are characterized by marked angular anisotropy and a comparatively weak dependence of the probability of the process—that is, the effective cross section for the process—on the particle’s energy. The nucleus formed as a result of a direct nuclear reaction is generally in either a weakly excited state or the ground state.

Direct nuclear reactions were discovered in the early 1950’s. The first to be detected were deuteron stripping (d, p) and pickup (p, d) reactions involving light nuclei. The protons and deuterons emitted in these reactions emerge primarily in the direction of the beam of incident particles. Direct nuclear reactions are known wherein a nucleon or group of nucleons crosses over from one of the colliding nuclei to another. A direct nuclear reaction of the type (x, xy) is called quasi-elastic scattering. In these reactions, the relation between the momenta and energies of the incident particle (x) and of the outgoing particles (x, y) is almost the same as in the elastic scattering of the particle x by the free particle y. The quasi-elastic scattering reactions produced by alpha particles, protons, and π-mesons in light nuclei are the best studied. Other reactions that have been observed include the knocking out of weakly bound particles—deuterons—from the nucleus, that is, reactions such as (p, pd).

Direct nuclear reactions are used to study the spectrum of nuclear levels and the structure of the nuclear periphery, especially peripheral correlated groups of nucleons (clusters). They are also used to obtain data on the interaction of unstable elementary particles with neutrons and nucleonic isobars; such data are important in studies of direct nuclear reactions at high energies.

38. Diffractive scattering - spetsifichiskoe elastic scattering of particles hadrons and nuclei, capable of absorbing the incident particles. Diffractive scattering of a wave nature and due to the fact that the absorption region distorts the wavefront of the incident wave on the system and leads to spread it into the geometric shadow. For small de Broglie wavelength of the particle (, where R – is the radius of the absorbing system, р – is momentum of the incident particle) дифракционное рассеяние аналогично дифракции света на непрозрачном экране. In the case of total absorption diffraction scattering is the only mechanism of elastic scattering. Characteristic angles at which diffraction scattering occurs, have a value (this follows from the uncertainty relation, as a scattering angle , where - change in the momentum of the particle in the direction perpendicular to the incident beam associated with the R ratio ). For scattering by a fully opaque sphere of radius R amplitudediffractive scattering angle υ and the differential cross section in the solid angle do respectively:

where - wavenumber, and  J₁(x) - Bessel function of the 1st order, which determines the characteristic oscillatory angular distribution . Cross section is concentrated mainly in the small-angle scattering i/kR and decreases rapidly to large υ. It is characterized by pronounced maxima and minima, coinciding with the extrema of the Bessel function. Diffractive scattering amplitude in this case is purely imaginary. Total cross sections and inelastic diffractive scattering processes do not depend on the energy and equal to each other, and the total cross section.

9. Delayed neutrons and their role in the regulation of the reactor. In nuclear engineering, a delayed neutron is a neutron emitted after a nuclear fission event, by one of the fission products (or actually, a fission product daughter after beta decay), any time from a few milliseconds to a few minutes after the fission event. Neutrons born within seconds of the fission are termed "prompt neutrons".In anuclear reactor large nuclides fission in two neutron-rich fission products (i.e. unstable nuclides). Many of these fission products then undergo radioactive decay (usually beta decay) and the resulting nuclides are left in an excited state. These usually immediately undergo gamma decay but a small fraction of them are excited enough to be able to decay by emitting a neutron in addition. The moment of beta decay of the precursor nuclides - which are the precursors of the delayed neutrons - happens orders of magnitude later compared to the emission of the prompt neutrons. Hence the neutron that originates from the precursor's decay is termed a delayed neutron. However, the "delay" in the neutron emission is due to the delay in beta decay, since neutron emission, like gamma emission, happens almost immediately after the beta decay. The various half lives of these decays that finally result in neutron emission, are thus the beta decay half lives of the precursor radionuclides.Delayed neutrons play an important role in nuclear reactor control and safety analysis. Delayed neutrons are associated with the beta decay of the fission products. After prompt fission neutron emission the residual fragments are still neutron rich and undergo a beta decay chain. The more neutron rich the fragment, the more energetic and faster the beta decay. In some cases the available energy in the beta decay is high enough to leave the residual nucleus in such a highly excited state that neutron emission instead of gamma emission occurs.Using U-235 as an example, this nucleus absorbs thermal neutrons, and the immediate mass products of a fission event are two large fission fragments, which are remnants of the formed U-236 nucleus. These fragments emit, on average, two or three free neutrons (in average 2.47), called "prompt" neutrons. A subsequent fission fragment occasionally undergoes a stage of radioactive decay (which is a beta minus decay) that yields a new nucleus (the precursor nucleus) in an excited state that emits an additional neutron, called a "delayed" neutron, to get to ground state. These neutron-emitting fission fragments are called delayed neutron precursor atoms.The delayed neutron fraction (DNF) is defined as:.These two factors, β and DNF, are not the same thing in case of a rapid change in the number of neutrons in the reactor.Another concept, is the effective fraction of delayed neutrons, which is the fraction of delayed neutrons weighted (over space, energy, and angle) on the adjoint neutron flux. This concept arises because delayed neutrons are emitted with an energy spectrum more thermalized relative to prompt neutrons. For low enriched uranium fuel working on a thermal neutron spectrum

27.The partial cross section. For a particle beam (say of neutrons, pions) incident on a target (liquid hydrogen), for each type of reaction in the scattering process labelled by an index r = 1, 2, 3..., it is calculated from:whereN is the number of target particles, illuminated by the beam containing n particles per unit volume in the beam (number density of particles) traveling with average velocity v in the rest frameof the target, and these two quantities combine into the flux of the beam J = nv. The cross section of the reaction is σr. Since the beam flux has dimensions of [length]−2·[time]−1 and σr has dimensions of [length]2 while N is a dimensionless number, the rate W has the dimensions of reciprocal time - which intuitively represents a frequency of recurring events.The above formula assumes the following: the beam particles all have the same kinetic energy,the number density of the beam particles is sufficiently low: allowing the interactions between the particles within the beam to be neglected,the number density of target particles is sufficiently low: so that only one scattering event per particle occurs as soon as the beam is incident with the target, and multiple scattering events within the target can be neglected,the de Broglie wavelength of the beam is much smaller than the inter-particle separations within the target, so that diffraction effects through the target can be neglected,the collision energy is sufficiently high allowing the binding energies in the target particles to be neglected.

These conditions are usually met in experiments, which allows for a very simple calculation of rate.Sometimes the rate per unit target particle, or rate density, is more useful. For reaction r:

36.Nuclear reactions proceeding through the compound nucleus. Either a low energy projectile is absorbed or a higher energy particle transfers energy to the nucleus, leaving it with too much energy to be fully bound together. On a time scale of about 10−19seconds, particles, usually neutrons, are "boiled" off. That is, it remains together until enough energy happens to be concentrated in one neutron to escape the mutual attraction. Charged particles rarely boil off because of the coulomb barrier. The excited quasi-bound nucleus is called a compound nucleus. Compound nucleus model was first formulated by Bohr. According to this model, the nuclear reaction occurs in two stages . In the first phase a particle and a target nucleus A form a coupled system of composite ( compound) core C, which in the second stage splits into core particle B and b: a + A C b + B.  The model is based on the assumption that the particle and , falling into the nucleus A interacts strongly with the nucleons . In the model of the compound nucleus is assumed that the mean free path of the incident particle is much smaller than the nucleus , whereby each particle enters the nucleus , it is captured . As a result of the incident particle and the nucleons of the nucleus excitation energy equal to a + Ba ( where a - the kinetic energy of the incident particle and , Ba - the binding energy of the particles in the nucleus and C) is evenly distributed between the nucleons in the nucleus , while the average excitation energy per nucleon equal (a + Ba) / A. if (a + Ba) / A << BN,where BN - nucleon binding energy in the compound nucleus C, it must pass a relatively long time compared with the time of flight of the particle through the nucleus equal to 2R / v, where v - velocity of the particle before on what or nucleon core focus enough energy to that he flew out of the nucleus

18.Nuclear reactions induced by protons. Nuclear reactions induced by protons . Interaction of protons with nuclei prevents the Coulomb barrier , so for light nuclei Nuclear reactions with protons are observed only since the proton energy xp order of several hundred keV , and for heavy nuclei - several MeV. For small xp main nuclear reactions - radiative capture of protons (p, v), and elastic ( p, p) and inelastic (p , p ") scattering of protons by nuclei. Light nuclei in the low probability of nuclear reactions xp is resonant . in medium and heavy nuclei , it reaches a significant value only in the energy region where no resonance structure . in the energy region xp, close to the height of the Coulomb barrier , there is a small number of excitation of isobaric analog states . nuclear reaction cross section is appreciable since 0.5 x0 (x0 - energy corresponding to the height of the Coulomb barrier ) and increases monotonically . Nuclear reactions (p, n) becomes predominant , if the compound nucleus excitation energy is sufficient for the emission of a neutron with an energy of 1 MeV ³ . Upon further increase xp final nucleus may have sufficient energy for emitting particles of a second . observed in this case the reaction (p, 2n) and (p, pn)

39. The kinematics of the two-particle nuclear reactions. Consider a + reaction A → b + B. We will use the non-relativistic approximation . Let A particle at rest in HP ac particle energy in Ta l.s.naletaet her. For energy hp Tb particles b, b emitted at an angle to the beam direction , the relation (see the following output ratios

where ma, mA, mb, mB - masses of particles a, A , b and B. Q -, energy of the reaction . If the second term under the square root is positive or zero, then the square root is taken "+" sign , while for b all values ​​from 0 to (see ris.k2a ) if negative, it is a given angle b to Tb or possibly two values ​​( segments AC1 and AC2, proportional to the energy of Tb in Fig. k2b ) , or none , and b values ​​scoped to the sharp corners , which root in the expression is validReaction energy Q, the angle of emission , energy and mass of the particles are related :

For the kinetic energy of the particle b in the cms T'b the relation :

Angles at the transition from the laboratory coordinate system in the center of mass is converted as follows: