Nuclear Reactions - Lecture Material

Lecture Material Nuclear Reactions Like radioactive decay, nuclear reactions obey the same conservation laws of total energy, momentum, total charge, mass number and spin. We will be primarily interested in reactions in which a projectile particle bombards a target nucleus. One of the first nuclear reactions studied was the bombardment of nitrogen with alpha particles. 14 7 N + 4 2 He 6 17 8 1 1 O + H This reaction can be symbolized using a shorthand notation in which the bombarding particle (α) and lighter product fragment (p) are written in parentheses between the target and product nuclei. For the reaction above this is written as: 14 N(α,p)17O When Q for this reaction is calculated by comparing the reactant mass with the product mass, it is found to be negative (-1.193 MeV). This means that there is some minimum energy the alpha particle must have in order for the reaction to occur. However, conservation of momentum requires that a portion [mp/(mt + mp)] of the bombarding particle energy is retained by the products as kinetic energy (in this case, ~4/18). Thus, the minimum energy needed for reaction equals Q(mt + mp)/mt where mt is the mass of the target and mp is the mass of the bombarding particle. For this reaction this is approximately equal to (-1.193 MeV)[(14 + 4)/14] = 1.5 MeV. If the projectile is positively charged, there will be a Coulomb barrier that must be surmounted so the projectile will require some additional kinetic energy for there to be interaction with the nucleus. The Coulomb barrier can be estimated from: t p c t p where Zt and Zp are the charges of the target and bombarding particle, respectively, and Rt and Rp are their radii in Fermi units. For the reaction above, the Coulomb barrier is approximately c 1/3 1/3 Thus, the minimum amount of energy needed to bring about appreciable reaction (threshold energy) for the reaction above is (3.6 MeV)[(14 + 4)/14] = 4.6 MeV. When the bombarding particle is a neutron there is no Coulomb barrier and neutrons of even very low energy can react with a nucleus. The most probable energy of a neutron population at 20EC is about 0.025 eV, and neutrons with this energy are referred to as thermal neutrons (nth). Such neutrons are found to have an especially high probability of reaction with a target nucleus. Cross Sections The probability of a nuclear reaction is generally expressed in terms of a cross section (σ) that has dimensions of area (cm2). The reaction probability is related to the likelihood of collision between the target and the bombarding particle which is related to the geometrical cross-sectional area of the target nucleus. Since nuclear areas are about 10-24 cm2, a cross section unit, the barn (b), has been defined: 1b = 1 x 10-24 cm2. The relationship between the reaction cross section and the energy of the bombarding particle is called an excitation function. Examples of excitation functions are given below. Neutron Absorption Excitation Function For Indium Notice that the neutron absorption cross section generally decreases with increasing neutron energy. This decrease is roughly proportional to 1/vn where vn is the neutron velocity. This 1/vn dependence relates to the wave properties of a neutron. The de Broglie wavelength of a neutron (λn) is given by n n where h is Planck’s constant and m is the neutron mass. The likelihood of capture (σcapt) % λn % 1/vn, thus the cross section is expected to decrease as the neutron energy increases. This also accounts for the very large capture cross sections (>100 b) that are sometimes found. The maximum reaction cross section is given by n 2 where R is the sum of the radii of the target and neutron. As the neutron energy increases (vn increases) λn decreases and the cross section approaches πR2. At higher neutron energies there are multiple peaks (resonance peaks). These very large values of the cross section are associated with the compound nucleus (target + captured neutron) being excited exactly to one of its discrete energy levels. The narrow width of these resonance peaks suggests that the compound nucleus lifetime is #10-14 s. The small spacing between peaks suggests that the excited compound nucleus must involve the excitation of many nucleons. Since the excitation energy of a compound nucleus formed by capture of a slow (thermal) neutron is only slightly higher than the binding energy of a neutron in the compound nucleus, it is unlikely that this energy would be concentrated on a neutron again to allow it to escape the nuclear potential well. Thus, de-excitation by gamma emission is much more likely and the main reaction involving thermal neutrons is the (n,γ) reaction. At higher neutron energies other reactions, including (n,n’), (n,2n), (n.α), (n,p) and (n,f), are possible. The excitation function for charged-particle reactions generally exhibits a threshold followed by an increase in the reaction cross section to some maximum value. The cross section then decreases as the threshold for some other more complex reaction is reached. Charged-particle reaction cross sections are usually much smaller (mb) than neutron absorption reaction cross sections. Cross sections are determined by measuring the amount of a radioactive product produced by a particular reaction. The simplest approach involves using thin targets for which the bombarding particle energy and intensity are not altered by the target. For a well-defined beam of particles striking a thin target, the number of reactions occurring in the target per unit time ( R) is given by where I is the number of incident particles per unit time; n is the number of target nuclei/cm3; x is the target thickness (cm); σ is the reaction cross section (cm2). If the target is irradiated by a uniform flux of particles incident on it from all directions, such as a sample irradiated with neutrons in a nuclear reactor, the rate of reaction is given by where N is the total number of target nuclei; φ is the flux density (particles/cm2 @ s); σ is the reaction cross section (cm2). In either case, it is assumed that the radioactive product does not decay significantly during the irradiation. In order to determine R to solve for the cross section, one must be able to determine the absolute disintegration rate of the radioactive product.