PHGN 422: Nuclear Physics Lecture 5: The Liquid Drop Model of the Nucleus
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1 PHGN 422: NUCLEAR PHYSICS PHGN 422: Nuclear Physics Lecture 5: The Liquid Drop Model of the Nucleus Prof. Kyle Leach September 5, 2017 Slide 1 KUgridlrcorner
2 Last Week... Nuclear binding results in a mass that is less than the sum of their constituent parts We can release that energy by performing nuclear reactions. The energy released is called the Q value. The atomic mass can be measured through several techniques, to a very high precision. Slide 2 Prof. Kyle Leach PHGN 422: Nuclear Physics
3 What Have We Learned About the Nucleus So Far? 1 The nuclear density is roughly constant for all nuclei Slide 3 Prof. Kyle Leach PHGN 422: Nuclear Physics
4 What Have We Learned About the Nucleus So Far? 1 The nuclear density is roughly constant for all nuclei 2 Nuclei are positively charged, and the nuclear charge density is also roughly constant Slide 3 Prof. Kyle Leach PHGN 422: Nuclear Physics
5 What Have We Learned About the Nucleus So Far? 1 The nuclear density is roughly constant for all nuclei 2 Nuclei are positively charged, and the nuclear charge density is also roughly constant 3 The strong force is attractive only at short range... Slide 3 Prof. Kyle Leach PHGN 422: Nuclear Physics
6 What Have We Learned About the Nucleus So Far? 1 The nuclear density is roughly constant for all nuclei 2 Nuclei are positively charged, and the nuclear charge density is also roughly constant 3 The strong force is attractive only at short range... 4 AND is repulsive at very short range (ie. nuclear matter is highly incompressible) Slide 3 Prof. Kyle Leach PHGN 422: Nuclear Physics
7 What Have We Learned About the Nucleus So Far? 1 The nuclear density is roughly constant for all nuclei 2 Nuclei are positively charged, and the nuclear charge density is also roughly constant 3 The strong force is attractive only at short range... 4 AND is repulsive at very short range (ie. nuclear matter is highly incompressible) These observations are remarkable, and have been performed with very simple concepts so far. We are now at the level of understanding where we can begin to theoretically model the nucleus in an attempt to predict our observations. Slide 3 Prof. Kyle Leach PHGN 422: Nuclear Physics
8 How Do We Begin to Understand Nuclear Binding? In order to understand nuclear binding, we need to derive a mathematical framework to explain our empirical observations. This method is typically referred to as mathematical modelling To start with, let us consider the nucleus as a charged drop of incompressible liquid. How do we do with that assumption? Slide 4 Prof. Kyle Leach PHGN 422: Nuclear Physics
9 The Liquid Drop Model A Simple Approach to Modelling the Atomic Nucleus The scattering experiments we saw previously suggested that nuclei have approximately constant density. We were then able to calculate the nuclear radius assuming a uniform sphere. A drop of uniform liquid has the same property. Slide 5 Prof. Kyle Leach PHGN 422: Nuclear Physics
10 The Liquid Drop Model A Simple Approach to Modelling the Atomic Nucleus The scattering experiments we saw previously suggested that nuclei have approximately constant density. We were then able to calculate the nuclear radius assuming a uniform sphere. A drop of uniform liquid has the same property. Source: Krane - Fig. 3.4 Slide 5 Prof. Kyle Leach PHGN 422: Nuclear Physics
11 The Liquid Drop Model A Simple Approach to Modelling the Atomic Nucleus The nuclear force is short-range, but does not allow for compression of nuclear matter. Molecules in a liquid drop have the same basic properties. Slide 6 Prof. Kyle Leach PHGN 422: Nuclear Physics
12 The Liquid Drop Model A Simple Approach to Modelling the Atomic Nucleus The nuclear force is short-range, but does not allow for compression of nuclear matter. Molecules in a liquid drop have the same basic properties. Slide 6 Prof. Kyle Leach PHGN 422: Nuclear Physics Source: Department of Chemistry, UC Davis
13 The Liquid Drop Model A Simple Approach to Modelling the Atomic Nucleus The nucleus is a positively charged object. For our purposes here, we can assume our liquid drop also has a uniform positive charge. Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
14 The Liquid Drop Model A Simple Approach to Modelling the Atomic Nucleus The nucleus is a positively charged object. For our purposes here, we can assume our liquid drop also has a uniform positive charge. Proton (π) + + Neutron (ν) + + Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics
15 The Liquid Drop Model A Simple Approach to Modelling the Atomic Nucleus We have been assuming spherical nuclei so far, but when additional energy is introduced into the system, nuclei can change their shape. A drop of liquid has the same property, and when other forces are present, it can deviate from a spherical shape. Slide 8 Prof. Kyle Leach PHGN 422: Nuclear Physics
16 The Liquid Drop Model A Simple Approach to Modelling the Atomic Nucleus We have been assuming spherical nuclei so far, but when additional energy is introduced into the system, nuclei can change their shape. A drop of liquid has the same property, and when other forces are present, it can deviate from a spherical shape. Source: L.P. Gaffney et al., Nature 497, (2013) Slide 8 Prof. Kyle Leach PHGN 422: Nuclear Physics
17 The Liquid Drop Model: The Volume Term Now that a suitable (conceptual) model has been proposed for how to treat the nucleus theoretically, we need to define the parameters for how we develop the mathematical formalism. We can start with our BE/A curve for nuclear matter: Slide 9 Prof. Kyle Leach PHGN 422: Nuclear Physics
18 The Liquid Drop Model: The Volume Term We ll discuss our starting point on the chalkboard... Slide 10 Prof. Kyle Leach PHGN 422: Nuclear Physics
19 The Liquid Drop Model: The Volume Term We ll discuss our starting point on the chalkboard... B volume = a V A Slide 10 Prof. Kyle Leach PHGN 422: Nuclear Physics
20 The Liquid Drop Model: The Volume Term We ll discuss our starting point on the chalkboard... B volume = a V A The volume term constant Slide 10 Prof. Kyle Leach PHGN 422: Nuclear Physics
21 The Liquid Drop Model: The Volume Term We ll discuss our starting point on the chalkboard... B volume = a V A The volume term constant Proportional to the number of nucleons Slide 10 Prof. Kyle Leach PHGN 422: Nuclear Physics
22 The Liquid Drop Model: The Volume Term We ll discuss our starting point on the chalkboard... B volume = a V A The volume term constant Proportional to the number of nucleons Empirically, a fit to the experimental data binding energies gives: a V = MeV!! But we know that B/A is roughly constant... B/A 8 MeV...so what is going on? Is our model that far off?...well sort of. Slide 10 Prof. Kyle Leach PHGN 422: Nuclear Physics
23 Corrections to our Leading Order Volume Approximation We already know that the liquid drop has further terms that define its binding energy other than accounting for just its volume of matter. Since the emperical value for a V is much greater than 8 MeV, we can surmise that each of these corrections lowers the total calculated binding energy. That means that our initial volume assumption is an overestimation of the total binding. Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics
24 Corrections to our Leading Order Volume Approximation We already know that the liquid drop has further terms that define its binding energy other than accounting for just its volume of matter. Since the emperical value for a V is much greater than 8 MeV, we can surmise that each of these corrections lowers the total calculated binding energy. That means that our initial volume assumption is an overestimation of the total binding. Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics
25 The Liquid Drop Model: The Surface Term First, we need to account for the fact that the nucleons on the surface have less neighbours, and do not exhibit the same binding as those in the interior (volume)... Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics
26 The Liquid Drop Model: The Surface Term First, we need to account for the fact that the nucleons on the surface have less neighbours, and do not exhibit the same binding as those in the interior (volume)... B surface = a S A 2/3 Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics
27 The Liquid Drop Model: The Surface Term First, we need to account for the fact that the nucleons on the surface have less neighbours, and do not exhibit the same binding as those in the interior (volume)... B surface = a S A 2/3 The surface term constant Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics
28 The Liquid Drop Model: The Surface Term First, we need to account for the fact that the nucleons on the surface have less neighbours, and do not exhibit the same binding as those in the interior (volume)... B surface = a S A 2/3 The surface term constant Proportional to A 2/3 Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics
29 The Liquid Drop Model: The Surface Term First, we need to account for the fact that the nucleons on the surface have less neighbours, and do not exhibit the same binding as those in the interior (volume)... B surface = a S A 2/3 The surface term constant Proportional to A 2/3 Empirically, a S = MeV Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics
30 The Liquid Drop Model: The Coulomb Term Protons in the nucleus repel each other due to their mutual positive charge, this reduces the binding energy further... Slide 13 Prof. Kyle Leach PHGN 422: Nuclear Physics
31 The Liquid Drop Model: The Coulomb Term Protons in the nucleus repel each other due to their mutual positive charge, this reduces the binding energy further... B Coulomb = a C Z(Z 1) A 1/3 Slide 13 Prof. Kyle Leach PHGN 422: Nuclear Physics
32 The Liquid Drop Model: The Coulomb Term Protons in the nucleus repel each other due to their mutual positive charge, this reduces the binding energy further... B Coulomb = a C Z(Z 1) A 1/3 The Coulomb term constant Slide 13 Prof. Kyle Leach PHGN 422: Nuclear Physics
33 The Liquid Drop Model: The Coulomb Term Protons in the nucleus repel each other due to their mutual positive charge, this reduces the binding energy further... B Coulomb = a C Z(Z 1) A 1/3 The Coulomb term constant Inversely proportional to A 1/3 Slide 13 Prof. Kyle Leach PHGN 422: Nuclear Physics
34 The Liquid Drop Model: The Coulomb Term Protons in the nucleus repel each other due to their mutual positive charge, this reduces the binding energy further... B Coulomb = a C Z(Z 1) A 1/3 The Coulomb term constant Inversely proportional to A 1/3 Empirically, a C = 0.71 MeV Slide 13 Prof. Kyle Leach PHGN 422: Nuclear Physics
35 Deviations from the Liquid Drop Analogy After accounting for the volume, surface, and Coulomb terms, how well have we done at our reproduction? Slide 14 Prof. Kyle Leach PHGN 422: Nuclear Physics
36 Deviations from the Liquid Drop Analogy After accounting for the volume, surface, and Coulomb terms, how well have we done at our reproduction? Slide 14 Prof. Kyle Leach PHGN 422: Nuclear Physics Source: Krane, Fig. 3.17
37 Neutron and Proton Numbers on the Nuclear Chart Slide 15 Prof. Kyle Leach PHGN 422: Nuclear Physics
38 The Liquid Drop Model: The Symmetry (or Asymmetry) Term For light nuclei, N Z (for heavy nuclei N is only slightly larger than Z). Where the Coulomb term would always favour Z = 0 for any A, we must account for the fact that nuclei are quantum objects (specifically that nucleons are fermions), and must obey the Pauli exclusion principle... Slide 16 Prof. Kyle Leach PHGN 422: Nuclear Physics
39 The Liquid Drop Model: The Symmetry (or Asymmetry) Term For light nuclei, N Z (for heavy nuclei N is only slightly larger than Z). Where the Coulomb term would always favour Z = 0 for any A, we must account for the fact that nuclei are quantum objects (specifically that nucleons are fermions), and must obey the Pauli exclusion principle... B asymmetry = a A (N Z)2 A Slide 16 Prof. Kyle Leach PHGN 422: Nuclear Physics
40 The Liquid Drop Model: The Symmetry (or Asymmetry) Term For light nuclei, N Z (for heavy nuclei N is only slightly larger than Z). Where the Coulomb term would always favour Z = 0 for any A, we must account for the fact that nuclei are quantum objects (specifically that nucleons are fermions), and must obey the Pauli exclusion principle... B asymmetry = a A (N Z)2 A The asymmetry term constant Slide 16 Prof. Kyle Leach PHGN 422: Nuclear Physics
41 The Liquid Drop Model: The Symmetry (or Asymmetry) Term For light nuclei, N Z (for heavy nuclei N is only slightly larger than Z). Where the Coulomb term would always favour Z = 0 for any A, we must account for the fact that nuclei are quantum objects (specifically that nucleons are fermions), and must obey the Pauli exclusion principle... B asymmetry = a A (N Z) 2 A The asymmetry term constant Inversely proportional to A Slide 16 Prof. Kyle Leach PHGN 422: Nuclear Physics
42 The Liquid Drop Model: The Symmetry (or Asymmetry) Term For light nuclei, N Z (for heavy nuclei N is only slightly larger than Z). Where the Coulomb term would always favour Z = 0 for any A, we must account for the fact that nuclei are quantum objects (specifically that nucleons are fermions), and must obey the Pauli exclusion principle... B asymmetry = a A (N Z) 2 A The asymmetry term constant Inversely proportional to A Empirically, a A = MeV Slide 16 Prof. Kyle Leach PHGN 422: Nuclear Physics
43 Emperical Observations on Nuclear Binding There is still one observation that can tell us something about the binding energy, and how nucleons interact with one another. How many nuclei with an even or odd number of protons and neutrons are stable? Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics
44 Emperical Observations on Nuclear Binding There is still one observation that can tell us something about the binding energy, and how nucleons interact with one another. How many nuclei with an even or odd number of protons and neutrons are stable? Z N Number of Stable Nuclei Even Even 177 Even Odd 58 Odd Even 54 Odd Odd 10 Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics
45 Emperical Observations on Nuclear Binding There is still one observation that can tell us something about the binding energy, and how nucleons interact with one another. How many nuclei with an even or odd number of protons and neutrons are stable? Z N Number of Stable Nuclei Even Even 177 Even Odd 58 Odd Even 54 Odd Odd 10 Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics
46 Emperical Observations on Nuclear Binding There is still one observation that can tell us something about the binding energy, and how nucleons interact with one another. How many nuclei with an even or odd number of protons and neutrons are stable? This suggests that there is a force we need to consider that adds additional binding when we have an even number of nucleons. We call this nuclear pairing Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics
47 The Liquid Drop Model: The Pairing Term We just saw that unpaired protons and neutrons are less bound. How do we represent this in our liquid drop model? +δ for even-even nuclei B pair = 0 for even-odd or odd-even δ for odd-odd nuclei Slide 18 Prof. Kyle Leach PHGN 422: Nuclear Physics
48 The Liquid Drop Model: The Pairing Term We just saw that unpaired protons and neutrons are less bound. How do we represent this in our liquid drop model? +δ for even-even nuclei B pair = 0 for even-odd or odd-even δ for odd-odd nuclei δ = a P A 1/2 Slide 18 Prof. Kyle Leach PHGN 422: Nuclear Physics
49 The Liquid Drop Model: The Pairing Term We just saw that unpaired protons and neutrons are less bound. How do we represent this in our liquid drop model? +δ for even-even nuclei B pair = 0 for even-odd or odd-even δ for odd-odd nuclei The pairing term constant δ = a P A 1/2 Slide 18 Prof. Kyle Leach PHGN 422: Nuclear Physics
50 The Liquid Drop Model: The Pairing Term We just saw that unpaired protons and neutrons are less bound. How do we represent this in our liquid drop model? +δ for even-even nuclei B pair = 0 for even-odd or odd-even δ for odd-odd nuclei δ = a P A 1/2 The pairing term constant Inversely proportional to A 1/2 Slide 18 Prof. Kyle Leach PHGN 422: Nuclear Physics
51 The Liquid Drop Model: The Pairing Term We just saw that unpaired protons and neutrons are less bound. How do we represent this in our liquid drop model? +δ for even-even nuclei B pair = 0 for even-odd or odd-even δ for odd-odd nuclei δ = a P A 1/2 The pairing term constant Inversely proportional to A 1/2 Empirically, a P = 12 MeV Slide 18 Prof. Kyle Leach PHGN 422: Nuclear Physics
52 Next Class... Reading Before Next Class Section 3.3 in Krane (if you haven t already) Next Class Topics Introduction of the Semi-Empirical Mass Formula Comparing our Liquid Drop Model to Experimental Observations (ie. how do we do?) The mass parabola, and energy concerns for radioactive decay. Assignment #1! Slide 19 Prof. Kyle Leach PHGN 422: Nuclear Physics
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