Cross Section

6 Key excerpts on "Cross Section"

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  eBook - ePub

  Nuclear Engineering Fundamentals

  A Practical Perspective

  • Robert E. Masterson(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  4

  Nuclear Cross Sections and Reaction Probabilities

  4.1 Nuclear Cross Sections and Their Uses

  When scientists first began to explore the structure of the atom, they were surprised to learn that the atom consisted primarily of empty space and that within this empty space there was a very small and extremely dense structure called the atomic nucleus , which was later discovered to have a diameter of about 10–12 cm (see Chapter 1 ). When they attempted to shoot other particles into the atom to break it apart, they found that the particles normally went right through the atom without hitting anything at all! Their experiments were equivalent to shooting a bullet into a haystack that has a small metal object (which is represented by the nucleus) inside of it. (In a real haystack, the nucleus would be about the size of a flea.) Most of the time, the bullet hits nothing but straw and passes completely through the haystack. However, occasionally it hits the metal object inside of it and scatters or ricochets away. Refer to Figure 4.1 to visualize the dynamics of this process. In the early days of the nuclear industry, scientists invented a way to measure the probability that an incoming particle (such as a neutron) would actually “hit” the nucleus and cause a nuclear reaction to occur. For this purpose, they invented the concept of the nuclear Cross Section.

  Essentially, a nuclear Cross Section is a convenient way of measuring the probability that an incoming particle will hit the nucleus (or another comparably sized object like an electron) and, as the result of that collision, cause a particular nuclear reaction or set of reactions to occur. There is a direct correlation between the size of a nuclear Cross Section and the cross-sectional area of the nucleus that the incident particle is trying to hit. A larger cross-sectional area means that there will be a higher probability for a reaction to occur, and a smaller Cross Sectional area means that the probability will be lower. The standard unit for measuring nuclear Cross Sections is a unit called the barn , (σ), which has a value of 10−24 cm2 . Hence a barn is roughly equivalent to the cross-sectional area of an atomic nucleus (i.e., A = π*D2 /4, where the diameter of the nucleus D is about 10–12 cm). (Refer to Figure 4.2 to get an idea of exactly how small a cross-sectional area of 10–24 cm can be.) In comparison, 1 Angstrom , a common unit for measuring the wavelength of light in the fields of chemistry and optics, is about 10–8 cm, and the atomic nucleus itself is about 1 femtometer (1 × 10–12 cm) in diameter. Hence the electron cloud in Figure 4.2 is about 1 Angstrom in diameter, while the nucleus is about 10,000 times smaller than this. Historically, the term “barn” was invented because it was believed that the early scientists who were shooting particles at the nucleus were having a hard time hitting the broad side of a barn

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  Data Analysis Techniques for Physical Scientists

  • Claude A. Pruneau(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  It is also common to report Cross Sections corresponding to specific particle production processes whether exclusive, semi- exclusive, or inclusive. One can, for example, measure the production Cross Section of pions, kaons, or other particle species, in the context of a specific interaction. By extension, one can also consider how such Cross Sections depend on the colliding partners (whether protons, neutrons, or nuclei, etc.) and the beam energy. It is also possible to break down Cross Section measurements in terms of other global collision characteristics such as the 392 Basic Measurements (estimated) impact parameter of collisions or the presence/absence of specific particles or other specific features in the final state of interactions. In closing this section, it is important to remark that the Cross Section of particular pro- cess is a property of that process and should thus be independent of experimental condi- tions. Evidently, experimental conditions may hinder or alter measurements of this Cross Section (integral or differential) and it is thus necessary to consider how experimental con- ditions may affect or alter the outcome of a specific measurement. This very important topic is discussed in detail in Chapter 12. 8.1.2 Relativistic Kinematics Modern experiments in particle and nuclear physics are commonly carried out at high collisional energies and involve measurements of particles with large momenta. The use of relativistic kinematics is thus usually advised, if not required. In this and the following sections, we use natural units: c = ¯ h = 1. In special relativity, and even more so in general relativity, one distinguishes vectors ac- cording to their properties of transformation when expressed in two different coordinate systems. Space–time coordinates are denoted with a contravariant four vector with compo- nents x μ : x μ = (x 0 , x 1 , x 2 , x 3 ) ≡ (t ,  x).

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  Neutron Cross Sections

  • Donald J. Hughes, R. A. Charpie, J. V. Dunworth(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  Many of these refer to specific interactions that are observed in limited energy bands only, such as phonon gain scattering or spin-dependent incoherent scattering and these we shall consider later, in appropriate chapters. But in spite of the variety, there are some Cross Section terms of rather general meaning that are useful throughout the entire energy range, and it is desirable to define these before proceeding. Certainly the most universally measured Cross Section is the total Cross Section, which as we have seen represents all the interaction processes that may result from the collision of a neutron with an atom, and is thus the sum of all the partial Cross Sections. Simple as this definition may seem, its interpretation, especially at low energy, requires careful attention. At energies above one eV or so, the individual atoms do not affect each other and the Cross Section of a particular atom is the same whether it is isolated or associated with other atoms in a crystal. At lower energy, however, the neutron waves scattered by different atoms interfere and the observed scattering, hence Cross Section, can be a strong function of the physical state of the material, for example whether it is crystalline or liquid. In this case, as in all Cross SectionS AND PRINCIPLES OF MEASUREMENT 7 others, the total Cross Section found in BNL 325 is obtained from Eq. (1-1), which means that it is averaged over the atoms as they exist in the sample and is not the Cross Section of an isolated atom. For exactness, we have referred in this paragraph to the atom, although most of the Cross Section is contributed by the nucleus, particularly at energies above one eV. There is an appreciable magnetic interaction between the neutron and the atomic electrons, as well as a very small electrostatic interaction, but these are observable only at low energy where the atomic electrons have an appreciable form factor.

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  Electron-Atom Collisions

  • Ian E. McCarthy, Erich Weigold(Authors)
  • 2005(Publication Date)
  • Cambridge University Press

    (Publisher)

  The development of electron—atom collision studies has also been strongly motivated by the need of data for testing and developing suitable theories of the scattering and collision processes, and for providing a tool for obtaining detailed information on the structure of the target atoms and molecules and final collision products. It has been aided by advances in vacuum techniques, sources of charged and neutral targets, progress in electron energy analysis and detection, progress in the development of electron sources (in particular the development of suitable sources of polarised electrons), the development of tuneable lasers, and the use of computers for online control of experiments and for data handling and analysis. Refinements in the experimental techniques have made it 2.1 Concept of Cross Sections 5 possible to study individual processes which have had to be averaged over in previous measurements. We cannot in this chapter make a comprehensive coverage of all of the experimental procedures presently being utilised. We will give a brief overview of the major modern techniques. The emphasis will be on differential cross-section measurements using single-collision beam—beam scattering geometry, since these are the most widely used techniques. They are versatile and also demonstrate most of the basic techniques involved in cross-section measurements. 2.1 Concept of Cross Sections The time-independent probability for the occurrence of a particular colli- sion process is represented by the corresponding scattering Cross Section. It characterises the scattering process and is well defined in most scattering experiments. There are some situations when a time-dependent probability must be considered and the normal definition of a scattering Cross Section is not applicable. 2.1.1 Differential Cross Section The effective interaction between an electron and an atom depends strongly on the electron velocity as well as the scattering angle and the nature of the process.

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  Gaseous Electronics

  Theory and Practice

  • Gorur Govinda Raju(Author)
  • 2005(Publication Date)
  • CRC Press

    (Publisher)

  We will not attempt an exhaustive treatment of experimental methods, but will restrict ourselves to those techniques that are relevant to our purposes. Earlier experimental techniques and results are extensively discussed in books by McDaniel, 1 Hasted, 2 Gilardini, 3 and Huxley and Crompton. 4 A review of techniques and results obtained up to the year 1971 is also given by Bederson and Kieffer. 5 A few general comments are, however, in order. 2.1 TOTAL COLLISION Cross SectionS The total collision Cross Section is the sum of elastic and all inelastic Cross Sections. The first Cross Sections to be measured were of this type because of the relative simplicity of the concept. The methodology of these experiments belonged to the category known as the transmission method. Total collision Cross Sections are measured using a number of different techniques that have been developed to improve accuracy and reduce discrepancies between the results obtained by different methods and theoretical analyses. The fundamental principle involved in these experiments is to measure the number of electrons that survive scattering or that get scattered as a result of collisions. Though some authors distinguish between these two techniques, the principle involved is not substantially different. A beam of electrons is generated by a suitable means such as photoelectric emission, 6 thermionic emission, 7,8 field emission, 2 and electron guns. 9 The electrons have low energy when emitted and are then accelerated through a set of grids to allow them to acquire the desired energy. Divergence of the beam is minimized by a set of electron lenses or the application of a magnetic field. 55 The electrons enter a collision chamber and are scattered as they pass through the chamber. The number of electrons scattered or the number that survive scattering is measured and related to the number entering the collision chamber to obtain the total scattering Cross Section.

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  An Introduction to the Physics of Interstellar Dust

  • Endrik Krugel(Author)
  • 2007(Publication Date)
  • CRC Press

    (Publisher)

  2 How to evaluate grain Cross Sections In section 2.1, we define Cross Sections, the most important quantities describ-ing the interaction between light and interstellar grains. Section 2.2 deals with the optical theorem which relates the intensity of light that is scattered by a particle into exactly the forward direction to its extinction Cross Section. In section 2.3 to 2.5, we learn how to compute the scattering and absorption coefficients of particles. Section 2.6 is concerned with a strange, but impor-tant property of the material constants that appear in Maxwell’s equations, like ε or µ . They are complex quantities and Kramers and Kronig discovered a dependence between the real and imaginary part. In the final section, we approximate the material constants of matter that is a mixture of different substances. 2.1 Defining Cross Sections 2.1.1 Cross Section for scattering, absorption and extinction For a single particle, the scattering Cross Section is defined as follows: Consider a plane monochromatic electromagnetic wave at frequency ν and with flux F 0 . The flux is the energy carried per unit time through a unit area and given in (1.44) as the absolute value of the Poynting vector. When the wave hits the particle, some light is scattered into the direction specified by the angles ( θ, φ ) as depicted in figure 2.1. The flux of this scattered light, F ( θ, φ ), when it is received at a large distance r from the particle, is obviously proportional to F 0 /r 2 ; we therefore write F ( θ, φ ) = F 0 k 2 r 2 · L ( θ, φ ) . (2.1) The function L ( θ, φ ) does not depend on r nor on F 0 . We have included in the denominator the wavenumber k = 2 π λ to make L ( θ, φ ) dimensionless; the wavelength λ is then the natural length unit to measure the distance r . 29 30 How to evaluate grain Cross Sections FIGURE 2.1 A grain scatters light from a plane wave with flux F 0 into the direction ( θ, φ ).

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