Nuclear Radius

5 Key excerpts on "Nuclear Radius"

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  eBook - ePub

  Nuclear and Radiochemistry

  Fundamentals and Applications

  • Jens-Volker Kratz(Author)
  • 2021(Publication Date)
  • Wiley-VCH

    (Publisher)

  4 Other Physical Properties of Nuclei

  4.1 Nuclear Radii

  We have already mentioned that nuclei have dimensions on the order of several femtometers: 1 fm = 10−15  m. Experiments designed to measure nuclear radii lead to the conclusion that, to a crude approximation at least, nuclear radii scale is

  (4.1)

  where r0 is a constant independent of A; that is, nuclear volumes are nearly proportional to the number of nucleons, and all nuclei have approximately the same density. Although nuclear densities (1014  g cm−3 ) are extremely high compared with ordinary matter, nuclei are by no means densely packed with nucleons. This is an important factor for the success of the single‐particle shell model. Different experimental methods lead to somewhat different values of r0 ranging between ≈1.1 and ≈1.6 fm and also differ in the degree to which their results are fitted by Eq. (4.1) . This is not surprising since different experiments measure quite different quantities. This can be the radius of the nuclear force field, the radius of the distribution of charges (protons), or the radius of the nuclear mass distribution. Measurements of the first two quantities have been available for a long time, while the third has become available only recently.

  The earliest information about nuclear sizes came from Rutherford scattering with α particles from natural radioactive sources. These showed that the distance of the closest approach D of the α particles was larger than the radius of the nuclear force field, for example, D turned out to be on the order of 10–20 fm for copper and 30–60 fm for uranium. With the advent of particle accelerators, strongly interacting particles could be brought into contact with the force field of target nuclei resulting in absorption at scattering angles Θ larger than the grazing angle Θgr . Classically, if the cross section (cross sections are introduced in Chapter 12 ) for elastic scattering is normalized to the Rutherford or Coulomb cross section, σel /σC  = 1 for small scattering angles and falls to zero at the grazing angle where absorption sets in. Due to the wave character of the ingoing and scattered particle, in reality, one observes an oscillatory structure of the quantal scattering cross section as a function of angle, which is reminiscent of optical diffraction patterns, see Figure 4.1

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  Halliday's Fundamentals of Physics, 1st Australian & New Zealand Edition

  • David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Nuclear radii A convenient unit for measuring distances on the scale of nuclei is the femtometre (= 10 −15 m). This unit is often called the fermi; the two names share the same abbreviation. Thus, 1 femtometer = 1 fermi = 1 fm = 10 -15 m. (42.2) We can learn about the size and structure of nuclei by bombarding them with a beam of high-energy electrons and observing how the nuclei deflect the incident electrons. The electrons must be energetic enough (at least 200 MeV) to have de Broglie wavelengths that are smaller than the nuclear structures they are to probe. The nucleus, like the atom, is not a solid object with a well-defined surface. Furthermore, although most nuclides are spherical, some are notably ellipsoidal. Nevertheless, electron-scattering experiments (as well as experiments of other kinds) allow us to assign to each nuclide an effective radius given by r = r 0 A 1∕3 , (42.3) in which A is the mass number and r 0 ≈ 1.2 fm. We see that the volume of a nucleus, which is proportional to r 3 , is directly proportional to the mass number A and is independent of the separate values of Z and N. That is, we can treat most nuclei as being a sphere with a volume that depends on the number of nucleons, regardless of their type. Equation 42.3 does not apply to halo nuclides, which are neutron-rich nuclides that were first produced in laboratories in the 1980s. These nuclides are larger than predicted by equation 42.3, because some of the neutrons form a halo around a spherical core of the protons and the rest of the neutrons. Lithium isotopes give an example. When a neutron is added to 8 Li to form 9 Li, neither of which are halo nuclides, the effective radius increases by about 4%. However, when two neutrons are added to 9 Li to form the neutron- rich isotope 11 Li (the largest of the lithium isotopes), they do not join that existing nucleus but instead form a halo around it, increasing the effective radius by about 30%.

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  Renewable Energy

  A First Course

  • Robert Ehrlich, Harold A. Geller, John R. Cressman(Authors)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  By using alpha particles of somewhat higher energies, which could approach the nucleus even closer, it is possible to actually measure the size of the nucleus and not merely set an upper limit. However, in practice, one usually uses electrons, rather than alpha particles in these experiments, since electrons have a strictly electromagnetic nuclear interaction, which is very well understood, and they do not feel the strong force. Such experiments have been conducted for many different target nuclei, and they show a striking regularity. For all the nuclei whose radii have been measured, the following simple dependence on mass number A holds: r = r 0 A 1 / 3, (3.10) where r 0 = 1.25 fm, although admittedly, it is a bit of a simplification to regard the nucleus as having a sharp well-defined surface. To understand the significance of this basic formula, simply calculate the volume of a spherical nucleus from its radius (Equation 3.10), and you will see that the volume is proportional to A. What does this fact imply? Note that this behavior (volume proportional to the number of particles) is quite different from the atom outside the nucleus, since the volume of an atom is certainly not proportional to the number of electrons it contains. Thus, unlike electrons, nucleons seem to behave like incompressible objects that are packed together in close proximity. Another way to express the situation is to note that all nuclei have precisely the same density, which, using Equation 3.10, is found to be the astonishing value of 2 × 10 17 kg/m 3 or 200 trillion times that of water. Does matter of such density exist anywhere in the universe, apart from the nucleus itself? The surprising answer is yes, inside of the strange astronomical objects known as neutron stars, which are the remnants of stars that have undergone supernova explosions toward the end of their lives. 2 2 A teaspoonful of nuclear matter would weigh 10 billion tons on Earth

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  Radioactivity

  History, Science, Vital Uses and Ominous Peril

  • Michael F. L'Annunziata(Author)
  • 2022(Publication Date)
  • Elsevier

    (Publisher)

  19.2. Nuclear Radius and density

  Rutherford utilized his discovery of the scattering of alpha particles by the atomic nucleus to determine the Nuclear Radius of aluminum. By selecting a metal foil of low Z (aluminum, Z =13) and thus low coulombic barrier to alpha penetration, and applying alpha particles of relatively high energy (7.7MeV) from a natural source whereby alpha particle scattering at an acute angle due to coulombic repulsion would begin to fail, Rutherford (1919, 1920a,b) was able to demonstrate that the distance of closest approach of these alpha particles to the atom center according to Coulomb's law was equivalent to the Nuclear Radius of aluminum, ∼5×10

  − 15

  m or 5fm (5 fermi).

  Since Rutherford's pioneering alpha particle scattering experiment, numerous nuclear scattering experiments have been carried out to measure the size of atomic nuclei, including electron scattering (Jansen et al., 1971 ; Sick, 1982 ), electron–proton scattering (Klarsfeld et al., 1986 ), electron–deuteron scattering (Sick and Trautmann, 1998 ), and high-energy beam interactions (Tanihata et al., 1985 ; Suzuki et al., 1999 ) and nuclear models (Li and Wang, 2021; Singh et al., 2020 ). The radius of an atomic nucleus obeys the general empirical formula

  (19.1)

  where r 0 is a constant, referred to as the radius parameter, and A is the mass number of the nucleus (Singh et al., 2020 ; Tolhoek and Brussaard, 1954 ; Elton, 1958 ; Angeli and Csatlós, 1977 ). A detailed derivation of Eq. (19.1) is provided by Elton (1958) . The value of the radius parameter r 0 , was confirmed by Royer (2008) to be the generally accepted value of 1.22fm with slight deviations according to nuclear models tested (Li and Wang, 2021). Obviously, the radius of the nucleus will increase with mass number. For a small nucleus, such as the nucleus of

  (equivalent to an alpha particle), the radius is calculated to be 1.9×10

  − 15

  m or 1.9fm. For larger nuclei, such as

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  Fundamentals of Physics, Extended

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  The Nuclear Force The force that controls the motions of atomic electrons is the familiar electro- magnetic force. To bind the nucleus together, however, there must be a strong attractive nuclear force of a totally different kind, strong enough to overcome the repulsive force between the (positively charged) nuclear protons and to bind both protons and neutrons into the tiny nuclear volume. The nuclear force must also be of short range because its influence does not extend very far beyond the nuclear “surface.” The present view is that the nuclear force that binds neutrons and protons in the nucleus is not a fundamental force of nature but is a secondary, or “spill- over,” effect of the strong force that binds quarks together to form neutrons and protons. In much the same way, the attractive force between certain neutral molecules is a spillover effect of the Coulomb electric force that acts within each molecule to bind it together. 1284 CHAPTER 42 NUCLEAR PHYSICS Figure 42-8 Energy levels for the nuclide 28 Al, deduced from nuclear reaction experiments. 0 1 2 3 28 Al Energy (MeV) 1285 42-2 SOME NUCLEAR PROPERTIES The radius r is given by Eq. 42-3 (r = r 0 A 1/3 ), where r 0 is 1.2 fm (= 1.2 × 10 –15 m). Substituting for r then leads to ρ = Am 4 3 πr 3 0 A = m 4 3 πr 3 0 . Note that A has canceled out; thus, this equation for density ρ applies to any nucleus that can be treated as spherical with a radius given by Eq. 42-3. Using 1.67 × 10 –27 kg for the mass m of a nucleon, we then have ρ = 1.67 × 10 −27 kg 4 3 π(1.2 × 10 −15 m) 3 ≈ 2 × 10 17 kg/m 3 . (Answer) This is about 2 × 10 14 times the density of water and is the density of neutron stars, which contain only neutrons. Sample Problem 42.03 Density of nuclear matter We can think of all nuclides as made up of a neutron – proton mixture that we can call nuclear matter. What is the density of nuclear matter? KEY IDEA We can find the (average) density ρ of a nucleus by dividing its total mass by its volume.

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