Differential Cross Section

10 Key excerpts on "Differential Cross Section"

• 

  eBook - PDF

  Quantum Mechanics

  A Modern Development

  • Leslie E Ballentine(Author)
  • 1998(Publication Date)
  • WSPC

    (Publisher)

  Chapter 16 Scattering The phenomenon of scattering was first mentioned in Sec. 2.1 of this book as an illustration of the fact that quantum mechanics does not predict the outcome of an individual measurement, but rather the statistical distribution or probabilities of all possible outcomes. Scattering is even more important than that illustration would indicate, much of our information about the interaction between particles being derived from scattering experiments. Entire books have been written on the subject of scattering theory, and this chapter will cover only the basic topics. 16.1 Cross Section The angular distribution of scattered particles in a particular process is described in terms of a Differential Cross Section. Suppose that a flux of Ji particles per unit area per unit time is incident on the target. The number of particles per unit time scattered into a narrow cone of solid angle df2, centered about the direction whose polar angles with respect to the incident flux are 0 and (j>, will be proportional to the incident flux J» and to the angular opening dft of the cone. Hence it may be written as J» ) dVt. The proportionality factor (7(0, ) is known as the Differential Cross Section. Suppose that a particle detector is located in the direction (0, (/>), at a suffi-ciently large distance r from the target so as to be outside of the incident beam. ► Fig. 16.1 Defining the Differential Cross Section [Eq. (16.1)]. 421 422 Ch. 16: Scattering If it subtends the solid angle dft it will receive J» ) dQ, scattered particles per unit time. Dividing this number by the area of the detector, we obtain the flux of scattered particles at the detector, J s = Jicr(9,)d£l/r 2 dQ. Thus the Differential Cross Section can be written as a(9,4>) = r -^-, (16.1) from which it is apparent that it has the dimensions of an area. Its value is independent of the distance r from the target to the detector because J s is inversely proportional to r 2 .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Quantum Mechanics

  • P. J. E. Peebles, P. James E. Peebles(Authors)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  The flux density in the scattered wave determines the differential scattering cross section, as follows. Differential Scattering Cross Section Consider a scattering experiment with an incident beam that contains n particles per unit volume moving toward a fixed target at speed v. The incident particle flux density is ]in = nv particles passing unit area normal to the beam in unit time. As indicated in Figure 45.1, a detector with area A normal to the scattered wave is placed at distance r from the interaction region in a direction 0,

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Phenomenology of Ultra-Relativistic Heavy-Ion Collisions

  • Wojciech Florkowski(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  PART IX SUPPLEMENTS This page intentionally left blank This page intentionally left blank Chapter 30 Cross Sections, Transition Rates, and Inclusive Distributions In this Chapter we collect the elementary information about the scattering theory . The basic quantities such as the cross section , the scattering matrix , and the tran-sition rate are defined. Following the popular treatment used in many textbooks, we consider the scattering processes in a simplified way, i.e., we put the colliding particles in a box and impose the periodic boundary conditions [1, 2]. This is done instead of the more realistic but also more involved analysis of the wave packets. The purpose of this Chapter is to present several issues that may be useful for better understanding of the ideas presented in the main parts of the book. For example, the expressions worked out below are used in Chap. 9, where the structure of the collision terms in the relativistic kinetic equations is analyzed. We start our presentation with a simple geometrical picture of the cross section that is introduced in Sec. 30.1. Then, we gradually come to more complex notions presented in Secs. 30.2, 30.3, and 30.4. Section 30.5 is devoted to the discussion of the inclusive cross sections . We show how the inclusive cross sections are related to the particle inclusive distributions . This issue is rarely discussed in textbooks but its good understanding is important for the correct interpretation of various high-energy measurements. 30.1 Geometric interpretation of cross section The scattering processes are commonly characterized by their cross sections. The value of the cross section, σ , determines the probability of a collision between the particles present in the two colliding beams. It may be interpreted as the effective transverse area of the colliding particles. To be more precise, let us assume that we have N 2 particles of the type “2”, which are at rest, v 2 = 0, in the volume V = dxdydz = Adz.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Electron-Atom Collisions

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

    (Publisher)

  Such cross sections are measured (e.g. Miiller-Fiedler, Jung and Ehrhardt, 1986; Goruganthu and Bonham, 1986; Oda, 1975) with instruments similar to those discussed in section 2.3 (e.g. fig. 2.5). A momentum selected projectile beam crosses a beam of target atoms at 90°, and the emitted electrons are analysed according to their energies and momenta. The double-Differential Cross Sections, which are discussed in chapter 10, show some general features. The fast scattered electrons are emitted into a narrow cone in the forward direction, with the angle of the cone being smaller for larger impact energies. The slow ejected electrons are emitted almost isotropically. The cross sections are normally put on an absolute scale by extrapolating the generalised oscillator strength for the incident electrons to zero momentum transfer (see section 2.2.2). Accurate absolute measurements of double-Differential Cross Sections are quite difficult to make. Measurements carefully taken by competent investigators often differ significantly. Kim (1983) gave a recommended 2.4 Ionisation 23 set of cross sections obtained after using various asymptotic boundary conditions and scaling laws to check the consistency of the available data. More recently Rudd (1991) proposed a different interpolation procedure to determine recommended cross sections for hydrogen and helium. 2.4.2 Single-Differential Cross Sections The single-Differential Cross Section is obtained by integrating the double- Differential Cross Section over all angles of emission of the electron M = I' (2.28) The form of this cross section is shown in fig. 2.7. It represents the energy loss spectrum integrated over all angles of the outgoing electrons. Due to the indistinguishability of these electrons, the single Differential Cross Section must be symmetric about (Et + JB /i )/2 = (£o — ^o)/2. The resonances shown in the continuum part of the spectrum in fig. 2.7 are schematic representations of autoionising transitions.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Neutron Cross Sections

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

    (Publisher)

  1-3, so that it measures only the scattered neutrons at a particular angle. In this way the Differential Cross Section is measured but, in contrast to the 12 NEUTRON CROSS SECTIONS total cross section, the absolute efficiency of the detector must usually be known. Furthermore, it must respond only to elastically scattered neutrons. The cross section cannot be obtained simply from a ratio of counting rates with the sample in place to an open beam because of the difference in shape and intensity of the direct and scattered neutron distributions. It is necessary to know , the solid angle subtended by the detector at the scattering sample, in order to evaluate the scattered flux, hence the differential scattering cross section, in absolute terms. Thus the neutron flux measured at the angle , relative to the incident flux, gives the differential elastic scattering cross section σ η (Ε; ): nv r 2 σ η (Ε; ) = T ^ — , (1-7) where r is the scatterer-detector distance and N' the total number of atoms in the scatterer. In a differential scattering measurement t must be sufficiently small so that there is little chance for neutrons to scatter more than once in the sample (i.e., negligible multiple scattering). Comparison of Eq. (1-7) for scattering with Eq. (1-5) shows that Eq. (1-7) contains the assumption that / is so small that there is negligible decrease in beam intensity in the sample (rnear unity). Only a small fraction of the beam is scattered, and only /4 of this fraction (if isotropic scattering) enters the detector, hence the lack of intensity in scattering measurements is easily appreciated. The elastic scattering cross section is determined from the integral over 4 solid angle of the Differential Cross Section, which, for isotropic scattering, is given simply as 4πσ η (Ε; ). Because of the complications mentioned, scattering cross sections in general are much more poorly known than total cross sections.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Atomic-Molecular Ionization by Electron Scattering

  Theory and Applications

  • K. N. Joshipura, Nigel Mason(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  T his chapter outlines the quantum scattering theories, employed later in this book, used for explaining the scattering of high energy (non-relativistic) electrons by atomic or molecular targets. The emphasis will be on a theoretical methodology based on the representation of the projectile-target system by a complex (optical) interaction potential. Electron scattering is measured (quantified) in terms of the ‘cross sections’ that result from the particular scattering event. As such, it is first necessary to define the different types of cross sections referred to, in this book, before further elaboration about the scattering processes. 2.1 DEFINITION OF ELECTRON SCATTERING CROSS SECTIONS The probability of a scattering process is expressed in terms of a ‘cross section’, which is measured typically using the unit cm 2 or more conveniently in Å 2 . The total cross section The total cross section, Q T (E i ), describes the probability of the incident electron interacting with the target in a collision at a specific energy E i and is defined by the total number of particles scattered in all directions per second per unit incident flux per scatterer. Q T (E i ) may be defined as the sum of individual cross sections for discrete scattering processes. Thus, Q T (E i ) = Q el (E i ) + Q inel (E i ) (2.1) where Q el (E i ) is the total elastic cross section and Q inel (E i ) is the summation of all inelastic scattering cross sections at incident energy E i . Furthermore, Q inel (E i ) = (2.2) where ∑Q exc is the summation of cross sections for electronic excitation (from an initial state i to a final state f ) and for molecules, also includes vibrational and rotational excitation. ∑Q ion is the summation of cross sections for all ionization processes. QUANTUM SCATTERING THEORIES 2 2 15 Quantum Scattering Theories The elastic cross section Elastic scattering conserves the total kinetic energy of the colliding particles.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Gaseous Electronics

  Theory and Practice

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

    (Publisher)

  In the early years of quantum mechanics Faxe´ n and Holtsmark 131 showed that the spherical part of may be split into partial waves according to their angular momentum ( L ) and that the scattering amplitude is given by f ð Þ ¼ 1 2 % ik X 1 i ¼ 0 2 L þ 1 ð Þ exp 2 i L ð Þ 1 ½ Š P L cos ð 3 : 18 Þ where L is the real phase shift and P L (cos ) is the L th Legendre polynomial. In theory the summation should be carried out for an infinite number of terms, but in practice the first few terms are the dominant terms and the remaining terms may be approximated by the effective range theory (ERT) 132 and Thompson’s expression for the contribution of the partial waves beyond a certain cutoff number L . The phase shift according to the ERT is L ¼ tan 1 2 % E 0 2 L þ 3 ð Þ 2 L þ 1 ð Þ 2 L 1 ð Þ ð 3 : 19 Þ where is the atomic polarizability in atomic units. Thompson’s expression for the scattering amplitude is given by f ð Þ ¼ 1 2 % ik X L i ¼ 0 2 L þ 1 ð Þ exp 2 i L ð Þ 1 ½ Š P L cos þ C L ð Þ ð 3 : 20 Þ where C L ð Þ ¼ % ak 1 3 1 2 sin 2 X L L ¼ 1 P L cos ð Þ 2 L þ 3 ð Þ 2 L 1 ð Þ ! ð 3 : 21 Þ Note that the upper limit for the summation in Equation 3.20 is L and not infinity. In helium the number of partial waves considered by Register and Trajmar 43 is 2 to 5, depending on the energy of the electron; higher energies require more terms. The Differential Cross Section is defined in quantum mechanics as dQ d ! ¼ f ð Þ 2 ð 3 : 22 Þ where f ( ) is the amplitude. The calculated Differential Cross Sections are then used for placing the measured cross sections below the first excitation potentials on an absolute scale. The total cross section is defined as 10 Q T ¼ 4 % k 2 X 1 0 2 L þ 1 ð Þ sin 2 L ð 3 : 23 Þ Data on Cross Sections—I. Rare Gases 121 and the momentum transfer cross section as Q M ¼ 4 % k 2 X 1 L ¼ 0 L þ 1 ð Þ sin 2 L L þ 1 ð Þ ð 3 : 24 Þ At very low energies only the lowest partial waves contribute to the scattering amplitude.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Asymptotic Diffraction Theory and Nuclear Scattering

  • Roy J. Glauber, Per Osland(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  While the most straightforward approaches have often been pro- ductive, they have tended to become lengthier with increasing particle energies, and to offer less direct insight into the meaning of even the more prominent features of the accumulated data. When the kinetic energies of the projectile particles begin to exceed the ener- gies of their interactions with target nuclei, the particles usually suffer only small deflections during collision processes. Since the particle wavelengths, at such ener- gies, are considerably smaller than nuclear dimensions, the angular distributions 3 4 Overview and Preview Figure 1.1 Differential Cross Section for elastic proton-lead scattering at 800 MeV [13]. of elastically scattered particles share many of the familiar properties of optical diffraction patterns. They tend to be strongly peaked for scattering in the forward direction, and to concentrate the scattered intensity within a fairly narrow cone. While the intensity tends to fall rapidly, on the average, with increasing scattering angle, it is also found often to oscillate more or less periodically. These features are quite evident, for example, in Fig. 1.1, which shows a succession of measured values of the Differential Cross Section for the elastic scattering of 800 MeV protons by 208 Pb [13]. The measurements, which intentionally omit the high intensity of Coulomb scattering at small angles, show a decrease of the Differential Cross Section by about seven orders of magnitude over the angular range from 3 ◦ to 30 ◦ . At the same time they show a fairly regular oscillation of the falling intensity which is surely a wave-mechanical interference effect of some sort. To make accurate measurements of scattering over such a large range of interac- tions has meant overcoming many experimental challenges [13]. But once they are overcome, the theoretical analysis of the results presents formidable problems as well, and these call for innovative solutions.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Automatic Detection Algorithms of Oil Spill in Radar Images

  • Maged Marghany(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  4

  Quantum Mechanical of Scattering Cross-Section Theory

  ABSTRACT: Scattering is the keystone to understand the mechanism of radar imaging. The conventional way of scattering is well addressed in many literature reviews. However, there is a new approach to comprehending the scattering mechanisms. This chapter delivers a new approach based on quantum mechanics. In this view, the Feynman concept for scattering is introduced. Moreover, the chapter has also demonstrated the dependence of the scattering theory on the spin. In this regard, this chapter delivers the correlations between scattering and wavefunction. For more details, the scattering theory is also tackled from the view of quantum particles.

  4.1      Definitions of Scattering

  Scattering is a general physical process where some forms of radiation, such as light, sound or moving particles, are forced to deviate from a straight trajectory by one or more path due to localized non-uniformities in the medium through which they pass. In this regard, the quantum mechanics describe the scattering as a function of atom-photon interaction. In this understanding, the definite detected targets are two incident photons with a certain energy, and two deviate photons with different energies. In this context, the probability of such a process (given the incident and the deviation) to occur is determined by the amplitude associated with this particular Feynman graph (Fig. 4.1 ). It is also worth noting that such a probability is not completely accurate since a single diagram is just one term of the infinite formal perturbative expansion providing the transition amplitude. However, it must not interpret the Feynman graph as a collision process. Figure 4.1 reveals that Positron 1 absorbs B and becomes positron 4, which emits C and becomes positron 3. Furthermore, electron 2 and positron 3 annihilate and producing D [52 , 53 , 54 , 55

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  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 ).

  Sign up to read

  Learn more about book