PHGN 422: Nuclear Physics Lecture 6: The Semi-Empirical Mass Formula

Transcription

1 PHGN 422: NUCLEAR PHYSICS PHGN 422: Nuclear Physics Lecture 6: The Semi-Empirical Mass Formula Prof. Kyle Leach September 7, 2017 Slide 1

2 Last Class... Assignment: Due September 22, 5pm to the TA mailbox in the Physics Office (Pines Trailer). To be fair to all students, late assignments will not be accepted...and the Pines trailer is usually locked at 5pm, so it will be obvious. Due to their similar properties, we are able to approximate the nucleus using an incompressible liquid drop This liquid drop model started with a volume term that overestimated the nuclear binding 4 additional terms were added based on emperical observations Now we can compile all of our terms to describe nuclear binding within this model... Slide 2 Prof. Kyle Leach PHGN 422: Nuclear Physics

3 BE from the Liquid Drop Model The Semi-Empirical Mass Formula where, B(A, Z) = a V A a S A 2/3 Z(Z 1) (A 2Z) 2 δ a C a A 1/3 A + a P A A, 1/2 +1 for even-even nuclei δ = 0 for even-odd or odd-even 1 for odd-odd nuclei Slide 3 Prof. Kyle Leach PHGN 422: Nuclear Physics

4 How Well Does Our BE Model Work? By comparing the predictions of our liquid drop model (the semi-empirical mass formula) to the measured mass data, we can get an idea of what effects in nuclei deviate from our simple assumptions. Slide 4 Prof. Kyle Leach PHGN 422: Nuclear Physics

5 PHGN 422: NUCLEAR PHYSICS How Well Does Our BE Model Work? By comparing the predictions of our liquid drop model (the semi-empirical mass formula) to the measured mass data, we can get an idea of what effects in nuclei deviate from our simple assumptions. Slide 4 Prof. Kyle Leach PHGN 422: Nuclear Physics

6 Small Deviations in the Binding Energy Now let us look at the trends of where our model deviates as a function of proton and neutron numbers, Z and N. What can this tell us about our assumptions of nuclear binding? Slide 5 Prof. Kyle Leach PHGN 422: Nuclear Physics

7 PHGN 422: NUCLEAR PHYSICS Small Deviations in the Binding Energy Now let us look at the trends of where our model deviates as a function of proton and neutron numbers, Z and N. What can this tell us about our assumptions of nuclear binding? Source: Heyde, Pg. 255 Slide 5 Prof. Kyle Leach PHGN 422: Nuclear Physics

8 Small Deviations in the Binding Energy Our experimental data seems to show that there are very well defined regions that have increased binding relative to our model Slide 6 Prof. Kyle Leach PHGN 422: Nuclear Physics

9 Small Deviations in the Binding Energy Our experimental data seems to show that there are very well defined regions that have increased binding relative to our model These regions also appear at the same nucleon number, regardless of the nucleon type (ie. protons or neutrons) Slide 6 Prof. Kyle Leach PHGN 422: Nuclear Physics

10 Small Deviations in the Binding Energy Our experimental data seems to show that there are very well defined regions that have increased binding relative to our model These regions also appear at the same nucleon number, regardless of the nucleon type (ie. protons or neutrons) These dips occur at N = Z = 2, 8, 20, 28, 50, 82, Slide 6 Prof. Kyle Leach PHGN 422: Nuclear Physics

11 Small Deviations in the Binding Energy Our experimental data seems to show that there are very well defined regions that have increased binding relative to our model These regions also appear at the same nucleon number, regardless of the nucleon type (ie. protons or neutrons) These dips occur at N = Z = 2, 8, 20, 28, 50, 82, We call these the Magic Numbers (seriously!), and it shows that there is an additional effect we have not accounted for. Ultimately, this points to the fact that our model has failed to describe nuclear binding from a microscopic level...but since we already knew that it was a severe approximation, we won t lose any sleep over that. Slide 6 Prof. Kyle Leach PHGN 422: Nuclear Physics

12 The Semi-Empirical Mass Formula Despite this fact, it is remarkable how close we have gotten to the experimental binding energies. What else can our simple formula tell us? The Semi-Empirical Mass Formula where, B(A, Z) = a V A a S A 2/3 Z(Z 1) (A 2Z) 2 δ a C a A 1/3 A + a P A A, 1/2 +1 for even-even nuclei δ = 0 for even-odd or odd-even 1 for odd-odd nuclei We ll go from binding energies to masses and investigate stability. To do that, we ll need some algebra and calculus, so let s head to the chalkboard... Slide 7 Prof. Kyle Leach PHGN 422: Nuclear Physics

13 The Mass Parabola and Stability Source: Krane, Fig Slide 8 Prof. Kyle Leach PHGN 422: Nuclear Physics

14 A = 125 Region of the Nuclear Chart Source: National Nuclear Data Center, Slide 9 Prof. Kyle Leach PHGN 422: Nuclear Physics

15 The Valley of Stability Source: The CERN Courier Slide 10 Prof. Kyle Leach PHGN 422: Nuclear Physics

16 Stable and Radioactive Nuclei Slide 11 Prof. Kyle Leach PHGN 422: Nuclear Physics Ruben Saakyan, Annu. Rev. Nucl. Part. Sci. 63, (2013)

17 Beta (β) and Double-Beta (ββ) Decay These decay modes represent the instability of this curve to first and second order. β Decay T 1/2 from a few ms to y Observed in > 1000 different nuclei from n to A 250 ββ Decay T 1/ y Observed in only 11 nuclei from 48 Ca to 238 U We will discuss β decay (along with the other forms of radioactive decay) in great detail in a couple of weeks. Slide 12 Prof. Kyle Leach PHGN 422: Nuclear Physics

18 Comment on Stability and the Mass Parabola By minimizing the semi-empirical mass formula as a function of the proton number, we found that it is possible for there to be multiple stable nuclei for a given A for even-a nuclei. Ruben Saakyan, Annu. Rev. Nucl. Part. Sci. 63, (2013) Slide 13 Prof. Kyle Leach PHGN 422: Nuclear Physics

19 Nuclear Stability and Natural Abundance This means that when we look at the periodic table, each element could contain one or more isotope that is stable. In physics we typically refer to the natural abundance as the fraction of each stable isotope for a given element as it is found on Earth. Slide 14 Prof. Kyle Leach PHGN 422: Nuclear Physics

20 Relative Natural Abundance of Uranium A T 1/2 Abundance y % y % y % When the Earth was younger, the isotopic composition of uranium was different. 1.7 billion years ago, 235 U was 3.1% relative abundant Slide 15 Prof. Kyle Leach PHGN 422: Nuclear Physics

21 One Last Comment on the Liquid Drop Model Using our very simple model of the liquid drop to describe nuclear binding as a function of A and Z, we have been able to discover Slide 16 Prof. Kyle Leach PHGN 422: Nuclear Physics

22 PHGN 422: NUCLEAR PHYSICS One Last Comment on the Liquid Drop Model Using our very simple model of the liquid drop to describe nuclear binding as a function of A and Z, we have been able to discover Source: Heyde, Pg. 255 Slide 16 Prof. Kyle Leach PHGN 422: Nuclear Physics

23 Stable and Radioactive Nuclei Across the Chart Phil Walker, New Scientist Magazine, October 2011 Slide 17 Prof. Kyle Leach PHGN 422: Nuclear Physics

24 Next Week... Reading Before Next Class Chapter 2, and Sections 3.4 and 3.5 in Krane Next Class Topics We will briefly discuss collective nuclear motions: angular momentum, dipole moments, quadrupole moments, etc.. Nuclear shapes and how they influence fission The mechanics of nuclear fission Slide 18 Prof. Kyle Leach PHGN 422: Nuclear Physics