PHGN 422: NUCLEAR PHYSICS PHGN 422: Nuclear Physics Lecture 3: Nuclear Radii, Masses, and Binding Energies Prof. Kyle Leach August 29, 2017 Slide 1
Last Week... The atomic nucleus is a very dense, positively charged object composed of protons and neutrons Nuclei are organized according to their Z and N values on the Nuclear Chart (or Chart of the Nuclides) Nuclei are held together by the strong interaction, and the nuclear force is attractive at short range, but repulsive at very short distances (we will talk about why today) So...back to our electron scattering experiments! Slide 2 Prof. Kyle Leach PHGN 422: Nuclear Physics
Electron Scattering on Nuclei Slide 3 Prof. Kyle Leach PHGN 422: Nuclear Physics Source: Fig. 3.1 (pg. 46) Introductory Nuclear Physics, Ken Krane
Light Scattering on an Opaque Object Source: Department of Physics, Brock University Slide 4 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Can This Tell Us About the Nucleus? Opaque Object Nuclear Matter Slide 5 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Nuclear Charge Distribution Source: Fig. 3.4 (pg. 49) Introductory Nuclear Physics, Ken Krane Slide 6 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Does This Tell Us About The Nucleus? 1 The boundary of the nucleus is not sharp, but displays a probability distribution Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Does This Tell Us About The Nucleus? 1 The boundary of the nucleus is not sharp, but displays a probability distribution The angular distributions from elastic scattering of electrons from nuclei do not show sharp minima Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Does This Tell Us About The Nucleus? 1 The boundary of the nucleus is not sharp, but displays a probability distribution The angular distributions from elastic scattering of electrons from nuclei do not show sharp minima These minima become even less sharp with increasing Z Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Does This Tell Us About The Nucleus? 1 The boundary of the nucleus is not sharp, but displays a probability distribution The angular distributions from elastic scattering of electrons from nuclei do not show sharp minima These minima become even less sharp with increasing Z 2 The central nuclear charge density is nearly the same for all nuclei Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Does This Tell Us About The Nucleus? 1 The boundary of the nucleus is not sharp, but displays a probability distribution The angular distributions from elastic scattering of electrons from nuclei do not show sharp minima These minima become even less sharp with increasing Z 2 The central nuclear charge density is nearly the same for all nuclei There is no dependence on the density of charge as a function of Z Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Does This Tell Us About The Nucleus? 1 The boundary of the nucleus is not sharp, but displays a probability distribution The angular distributions from elastic scattering of electrons from nuclei do not show sharp minima These minima become even less sharp with increasing Z 2 The central nuclear charge density is nearly the same for all nuclei There is no dependence on the density of charge as a function of Z Nucleons do not seem to preferentially organize based on type (ie. protons or neutrons) Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Does This Tell Us About The Nucleus? 1 The boundary of the nucleus is not sharp, but displays a probability distribution The angular distributions from elastic scattering of electrons from nuclei do not show sharp minima These minima become even less sharp with increasing Z 2 The central nuclear charge density is nearly the same for all nuclei There is no dependence on the density of charge as a function of Z Nucleons do not seem to preferentially organize based on type (ie. protons or neutrons) 3 The overall matter density of all nucleons in the nucleus must therefore be constant as well? (number of nucleons per unit volume) Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Does This Tell Us About The Nucleus? 1 The boundary of the nucleus is not sharp, but displays a probability distribution The angular distributions from elastic scattering of electrons from nuclei do not show sharp minima These minima become even less sharp with increasing Z 2 The central nuclear charge density is nearly the same for all nuclei There is no dependence on the density of charge as a function of Z Nucleons do not seem to preferentially organize based on type (ie. protons or neutrons) 3 The overall matter density of all nucleons in the nucleus must therefore be constant as well? (number of nucleons per unit volume) If this is true, we should be able to determine what the density of nuclear matter is Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
What Does This Tell Us About The Nucleus? 1 The boundary of the nucleus is not sharp, but displays a probability distribution The angular distributions from elastic scattering of electrons from nuclei do not show sharp minima These minima become even less sharp with increasing Z 2 The central nuclear charge density is nearly the same for all nuclei There is no dependence on the density of charge as a function of Z Nucleons do not seem to preferentially organize based on type (ie. protons or neutrons) 3 The overall matter density of all nucleons in the nucleus must therefore be constant as well? (number of nucleons per unit volume) If this is true, we should be able to determine what the density of nuclear matter is Also, can we find a generic way of obtaining the matter radius of a given nucleus? Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
Density of Nuclear Matter Well, to start...let s assume that the nucleus is a perfect sphere. From here, we can estimate the volume and perhaps the density... + + Proton (π) + Neutron (ν) + V = 4 3 πr3 Slide 8 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Nuclear Matter Radius If the nuclear matter density is also indeed constant for all nuclei: V = 4 3 πr3 constant Then, we can relate the radius of a nucleus to the number of nucleons A: R A 1/3 To determine this proportionality constant, we can relate the total nuclear matter radius R to the matter radius of the individual nucleons R 0 Slide 9 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Nuclear Matter Radius The nucleons can also be considered spherical: Therefore: 4 3 πr3 = A 4 3 πr3 0 = R = R 0 A 1/3 Experimentally we know that R 0 1.2 fm. So, the nuclear matter radius is R = 1.2 A 1/3! Further detailed discussion on this topic can be found in Chapter 3.1 of Krane. Slide 10 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Nature of Nuclear Matter One of the most remarkable conclusions from all of this is that nuclear matter does not seem to change density regardless of the size of the nucleus!! In other words, the number of nucleons per unit of volume is roughly constant for all nuclei. How dense is nuclear matter (comparatively speaking). Well... Sea Water: 1.0 10 3 kg/m 3 Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Nature of Nuclear Matter One of the most remarkable conclusions from all of this is that nuclear matter does not seem to change density regardless of the size of the nucleus!! In other words, the number of nucleons per unit of volume is roughly constant for all nuclei. How dense is nuclear matter (comparatively speaking). Well... Sea Water: 1.0 10 3 kg/m 3 Tin Oxide: 1.6 10 3 kg/m 3 Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Nature of Nuclear Matter One of the most remarkable conclusions from all of this is that nuclear matter does not seem to change density regardless of the size of the nucleus!! In other words, the number of nucleons per unit of volume is roughly constant for all nuclei. How dense is nuclear matter (comparatively speaking). Well... Sea Water: 1.0 10 3 kg/m 3 Tin Oxide: 1.6 10 3 kg/m 3 Steel: 1.1 10 4 kg/m 3 Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Nature of Nuclear Matter One of the most remarkable conclusions from all of this is that nuclear matter does not seem to change density regardless of the size of the nucleus!! In other words, the number of nucleons per unit of volume is roughly constant for all nuclei. How dense is nuclear matter (comparatively speaking). Well... Sea Water: 1.0 10 3 kg/m 3 Tin Oxide: 1.6 10 3 kg/m 3 Steel: 1.1 10 4 kg/m 3 Lead: 2.5 10 4 kg/m 3 Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Nature of Nuclear Matter One of the most remarkable conclusions from all of this is that nuclear matter does not seem to change density regardless of the size of the nucleus!! In other words, the number of nucleons per unit of volume is roughly constant for all nuclei. How dense is nuclear matter (comparatively speaking). Well... Sea Water: 1.0 10 3 kg/m 3 Tin Oxide: 1.6 10 3 kg/m 3 Steel: 1.1 10 4 kg/m 3 Lead: 2.5 10 4 kg/m 3 Core of the Sun: 1.5 10 5 kg/m 3 Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Nature of Nuclear Matter One of the most remarkable conclusions from all of this is that nuclear matter does not seem to change density regardless of the size of the nucleus!! In other words, the number of nucleons per unit of volume is roughly constant for all nuclei. How dense is nuclear matter (comparatively speaking). Well... Sea Water: 1.0 10 3 kg/m 3 Tin Oxide: 1.6 10 3 kg/m 3 Steel: 1.1 10 4 kg/m 3 Lead: 2.5 10 4 kg/m 3 Core of the Sun: 1.5 10 5 kg/m 3 Nuclear Matter: 2.3 10 17 kg/m 3 Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics
Question: What if the nucleus were nearly 20 orders of magnitude larger? Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics
PHGN 422: NUCLEAR PHYSICS Question: What if the nucleus were nearly 20 orders of magnitude larger? Well, this is not hypothetical...these are known as neutron stars Source: NASA.gov Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics
PHGN 422: NUCLEAR PHYSICS Question: What if the nucleus were nearly 20 orders of magnitude larger? Well, this is not hypothetical...these are known as neutron stars Source: NASA.gov Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics
Bound Nuclear Systems Limits of Nuclear Existence Putting aside neutron stars for now, let us take a look at the limits of what nuclei can exist, and how we define it. Slide 13 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Atomic Mass and Nuclear Binding Energy Slide 14 Prof. Kyle Leach PHGN 422: Nuclear Physics
The Atomic Mass and Nuclear Binding Energy As we briefly mentioned last week, the mass of a given atom is not simply the sum of neutron, proton, and electron masses, ie: ( ) Z M( A ZX N )c 2 Z m p c 2 + N m n c 2 Z m e c 2 B i i=1 For a nucleus to exist (ie. be a bound system), the following constraint must be satisfied (neglecting the electrons for a moment): M( A ZX N )c 2 < Z m p c 2 + N m n c 2 For the nucleons to be bound inside of the nucleus, there needs to be some energy difference. We call this the Binding Energy. We ll define what we mean on the chalkboard... Slide 15 Prof. Kyle Leach PHGN 422: Nuclear Physics
Mass Excess Since the atomic mass in MeV/c 2 can become a cumbersome way of dealing with larger nuclei (ie. m( 208 Pb) = 193 733 MeV/c 2 ) We can define a useful experimental mass value relative to our definition of the atomic mass unit in Lecture 1 (1u = 931.502 MeV/c 2 ). m( A ZX N )c 2 = (A u)c 2 + c 2 = c 2 = m( A ZX N )c 2 A) u Where is referred to as the Mass Excess or Mass Defect, and helps us to quantify how much a specific nucleus deviates from our approximation of the atomic mass unit. It can be either positive or negative, as long as we satisfy M( A Z X N)c 2 < Z m p c 2 + N m n c 2. Slide 16 Prof. Kyle Leach PHGN 422: Nuclear Physics
Example: What is the Mass Excess ( ) for 16 O in MeV? First we ll start with the experimentally measured mass of 16 O in u: Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics
Example: What is the Mass Excess ( ) for 16 O in MeV? First we ll start with the experimentally measured mass of 16 O in u: Remember, we say mass 16 for 16 O, but this is not exactly true. Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics
Example: What is the Mass Excess ( ) for 16 O in MeV? First we ll start with the experimentally measured mass of 16 O in u: Remember, we say mass 16 for 16 O, but this is not exactly true. m( A ZX N )c 2 = 15.994915 u Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics
Example: What is the Mass Excess ( ) for 16 O in MeV? First we ll start with the experimentally measured mass of 16 O in u: Remember, we say mass 16 for 16 O, but this is not exactly true. m( A ZX N )c 2 = 15.994915 u Now solve for the mass excess, (recall 1 u = 931.505 MeV/c 2 ) = m( A ZX N A) u = (15.994915 16) 931.505 MeV = 4.737 MeV Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics
Characteristics of Nuclear Binding The Proton and Neutron Separation Energies (S p and S n ) Analogous to atomic ionization energies, these separation energies can tell us about the binding strength for an individual nucleon. We can define these on the chalkboard: Slide 18 Prof. Kyle Leach PHGN 422: Nuclear Physics
Characteristics of Nuclear Binding The Proton and Neutron Separation Energies (S p and S n ) Analogous to atomic ionization energies, these separation energies can tell us about the binding strength for an individual nucleon. We can define these on the chalkboard: We can also look at the trends of how nuclear binding changes as a function of the mass number A Slide 18 Prof. Kyle Leach PHGN 422: Nuclear Physics
Characteristics of Nuclear Binding Binding Energy per Nucleon (BE/A) Slide 19 Prof. Kyle Leach PHGN 422: Nuclear Physics
Characteristics of Nuclear Binding Binding Energy per Nucleon (BE/A) This brings us to some other revelations about the way nuclei behave: 1 Most nuclei have almost exactly the same BE/A, which is roughly 8 MeV/A. This means the nuclear force saturates such that only each nucleon can interact with a few of its neighbours. Recall that the nuclear force is strongly attractive ONLY at short distances ( 1 fm). Slide 20 Prof. Kyle Leach PHGN 422: Nuclear Physics
Characteristics of Nuclear Binding Binding Energy per Nucleon (BE/A) This brings us to some other revelations about the way nuclei behave: 1 Most nuclei have almost exactly the same BE/A, which is roughly 8 MeV/A. This means the nuclear force saturates such that only each nucleon can interact with a few of its neighbours. Recall that the nuclear force is strongly attractive ONLY at short distances ( 1 fm). 2 The most bound nuclei are in the region of A 56 62 Slide 20 Prof. Kyle Leach PHGN 422: Nuclear Physics
Characteristics of Nuclear Binding Binding Energy per Nucleon (BE/A) This brings us to some other revelations about the way nuclei behave: 1 Most nuclei have almost exactly the same BE/A, which is roughly 8 MeV/A. This means the nuclear force saturates such that only each nucleon can interact with a few of its neighbours. Recall that the nuclear force is strongly attractive ONLY at short distances ( 1 fm). 2 The most bound nuclei are in the region of A 56 62 3 Some structure in this curve also exists (particularly for 4 He) that results from quantum effects of the nucleus. We will discuss the shell structure of nuclei in a couple of weeks. Slide 20 Prof. Kyle Leach PHGN 422: Nuclear Physics
Characteristics of Nuclear Binding Binding Energy per Nucleon (BE/A) This brings us to some other revelations about the way nuclei behave: 1 Most nuclei have almost exactly the same BE/A, which is roughly 8 MeV/A. This means the nuclear force saturates such that only each nucleon can interact with a few of its neighbours. Recall that the nuclear force is strongly attractive ONLY at short distances ( 1 fm). 2 The most bound nuclei are in the region of A 56 62 3 Some structure in this curve also exists (particularly for 4 He) that results from quantum effects of the nucleus. We will discuss the shell structure of nuclei in a couple of weeks. 4 Nuclei on the left of the peak can release energy by joining together (Nuclear Fusion) Slide 20 Prof. Kyle Leach PHGN 422: Nuclear Physics
Characteristics of Nuclear Binding Binding Energy per Nucleon (BE/A) This brings us to some other revelations about the way nuclei behave: 1 Most nuclei have almost exactly the same BE/A, which is roughly 8 MeV/A. This means the nuclear force saturates such that only each nucleon can interact with a few of its neighbours. Recall that the nuclear force is strongly attractive ONLY at short distances ( 1 fm). 2 The most bound nuclei are in the region of A 56 62 3 Some structure in this curve also exists (particularly for 4 He) that results from quantum effects of the nucleus. We will discuss the shell structure of nuclei in a couple of weeks. 4 Nuclei on the left of the peak can release energy by joining together (Nuclear Fusion) 5 Nuclei on the right of the peak can release energy by breaking apart (Nuclear Fission) Slide 20 Prof. Kyle Leach PHGN 422: Nuclear Physics
Characteristics of Nuclear Binding Binding Energy per Nucleon (BE/A) Source: The Open University Slide 21 Prof. Kyle Leach PHGN 422: Nuclear Physics
PHGN 422: NUCLEAR PHYSICS Nuclear Fusion in Stars Source: A.C. Phillips, The Physics of Stars, 2nd Edition (Wiley, 1999) Slide 22 Prof. Kyle Leach PHGN 422: Nuclear Physics
Nuclear Fission in Reactors Source: Department of Physics, UC Davis Slide 23 Prof. Kyle Leach PHGN 422: Nuclear Physics
Next Class... Reading Before Next Class Sections 3.2 and 3.3 (first part) in Krane More on binding energy Next Class Topics Ways to release of energy in a nuclear decay or reaction The experimental determination of atomic masses...and why do we care? Slide 24 Prof. Kyle Leach PHGN 422: Nuclear Physics