Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 10

The semi-empirical mass formula In nuclear physics, the semi-empirical mass formula is a formula for calculating the approximate nuclear binding energy \(B\) of an atomic nucleus with atomic number \(Z\) and mass number \(A\) : $$ B=a_{1} A-a_{2} A^{2 / 3}-a_{3} \frac{Z^{2}}{A^{1 / 3}}-a_{4} \frac{(A-2 Z)^{2}}{A}+\frac{a_{5}}{A^{1 / 2}} $$ where, in units of millions of electron volts, the constants are \(a_{1}=15.8, a_{2}=18.3\), \(a_{3}=0.714, a_{4}=23.2\), and $$ a_{5}= \begin{cases}0 & \text { if } A \text { is odd, } \\ 12.0 & \text { if } A \text { and } Z \text { are both even, } \\ -12.0 & \text { if } A \text { is even and } Z \text { is odd. }\end{cases} $$ a) Write a program that takes as its input the values of \(A\) and \(Z\), and prints out the binding energy for the corresponding atom. Use your program to find the binding energy of an atom with \(A=58\) and \(Z=28\). (Hint: The correct answer is around \(500 \mathrm{MeV}\) ) b) Modify your program to print out not the total binding energy B, but the binding energy per nucleon, which is \(B / A\). c) Now modify your program so that it takes as input just a single value of the atomic number \(Z\) and then goes through all values of \(A\) from \(A=Z\) to \(A=3 Z\), to find the one that has the largest binding energy per nucleon. This is the most stable nucleus with the given atomic number. Have your program print out the value of \(A\) for this most stable nucleus and the value of the binding energy per nucleon d) Modify your program again so that, instead of taking \(Z\) as input, it runs through all values of \(Z\) from 1 to 100 and prints out the most stable value of \(A\) for each one. At what value of \(Z\) does the maximum binding energy per nucleon occur? (The true answer, in real life, is \(Z=28\), which is nickel.)

Short Answer

Expert verified
Implement a function to compute binding energy B for A and Z. Modify it to find B/A, then to find the most stable nucleus for a given Z, and finally to check stability for all Z from 1 to 100.

Step by step solution

01

- Understand the semi-empirical mass formula

The semi-empirical mass formula is used to calculate the binding energy (B) of an atomic nucleus with atomic number Z and mass number A. The formula includes several constants which are provided and depend on whether A and Z are odd or even.
02

- Write the function to calculate B

Create a function that takes A and Z as inputs and calculates the binding energy B using the given formula. You'll need to handle the different cases for the constant a5 based on whether both A and Z are even, A is odd, or A is even and Z is odd.
03

- Implement the program

Write a program that calls the function developed in Step 2 and prints out the binding energy B for given values of A and Z. Use the program to find the binding energy for A=58 and Z=28.
04

- Modify program to calculate B/A

Change the existing program to print the binding energy per nucleon, which is B divided by A.
05

- Find the most stable nucleus for a given Z

Modify the program so that it takes Z as input and iterates from A=Z to A=3Z to find the value of A that results in the maximum binding energy per nucleon. Print the value of A and the corresponding binding energy per nucleon.
06

- Iterate Z from 1 to 100

Extend the program to run through all values of Z from 1 to 100, find the most stable value of A for each Z, print the most stable value of A, and the corresponding binding energy per nucleon. Determine the Z value for which the maximum binding energy per nucleon occurs.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nuclear Binding Energy
Nuclear binding energy is a key concept in understanding the stability of atomic nuclei. It represents the energy needed to disassemble a nucleus into its constituent protons and neutrons.
Nuclei are bound together by the strong nuclear force, overcoming the repulsive electrostatic forces between protons. This strong nuclear force is much more powerful than the repulsive electromagnetic force, but only acts over a very short range.
The semi-empirical mass formula provides a way to calculate the binding energy of a nucleus based on its atomic number (Z) and mass number (A). This formula incorporates both attractive and repulsive forces within the nucleus.
Key points to remember about nuclear binding energy:
  • Higher binding energy generally means a more stable nucleus.
  • Nuclear binding energy is quantified in MeV (mega-electronvolts).
  • Binding energy per nucleon often provides a useful measure of nuclear stability.
  • The semi-empirical mass formula balances volume, surface, Coulomb, asymmetry, and pairing terms to estimate binding energy.
Atomic Nucleus
The atomic nucleus is the core of an atom, composed of protons and neutrons, collectively called nucleons.
The number of protons determines the element's atomic number (Z), while the total number of protons and neutrons defines the mass number (A).
The nucleus is held together by the nuclear force, which is short-ranged but extremely powerful, especially when compared to the electromagnetic force.
Understanding the structure of the atomic nucleus involves knowing about:
  • The arrangement and number of protons and neutrons.
  • The types of forces acting within the nucleus.
  • The role binding energy plays in nuclear stability.
Atomic nuclei can vary significantly in their stability. Typically, elements with a balanced number of protons and neutrons are more stable. However, isotopes—variations of elements with different numbers of neutrons—can offer rich insights into nuclear physics.
The semi-empirical mass formula helps predict how different factors influence the nucleus’s stability by quantifying their contributions to nuclear binding energy.
Programming in Physics
Programming plays an essential role in modern physics, enabling physicists to model complex systems, analyze data, and solve intricate equations numerically.
In our example problem, we use programming to compute nuclear binding energies using the semi-empirical mass formula. Here are key aspects of programming in this context:
  • Using functions to encapsulate the calculations, ensuring reusability and modularity.
  • Handling different cases based on whether values of A and Z are odd or even.
  • Iterating over ranges of A and Z to find stable nuclei efficiently.
  • Displaying results in a clear and concise manner.
  • Ensuring that the program can handle various inputs dynamically.
Coding such problems involves translating physical concepts into mathematical expressions that a computer can work with.
By iterating over a range of values and calculating the binding energies, we can determine the most stable nuclei for given atomic numbers, showcasing the power of computational methods in physics.
To implement a fully functional program, one might use a language like Python due to its readability and strong scientific libraries.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Quantum potential step A well-known quantum mechanics problem involves a particle of mass \(m\) that encounters a one-dimensional potential step, like this: The particle with initial kinetic energy \(E\) and wavevector \(k_{1}=\sqrt{2 m E} / \hbar\) enters from the left and encounters a sudden jump in potential energy of height \(V\) at position \(x=0\). By solving the Schrödinger equation, one can show that when \(E>V\) the particle may either (a) pass the step, in which case it has a lower kinetic energy of \(E-V\) on the other side and a correspondingly smaller wavevector of \(k_{2}=\sqrt{2 m(E-V)} / \hbar\), or \((b)\) it may be reflected, keeping all of its kinetic energy and an unchanged wavevector but moving in the opposite direction. The probabilities \(T\) and \(R\) for transmission and reflection are given by $$ T=\frac{4 k_{1} k_{2}}{\left(k_{1}+k_{2}\right)^{2}}, \quad R=\left(\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right)^{2} . $$ Suppose we have a particle with mass equal to the electron mass \(m=9.11 \times\) \(10^{-31} \mathrm{~kg}\) and energy \(10 \mathrm{eV}\) encountering a potential step of height \(9 \mathrm{eV}\). Write a Python program to compute and print out the transmission and reflection probabilities using the formulas above.

The Madelung constant In condensed matter physics the Madelung constant gives the total electric potential felt by an atom in a solid. It depends on the charges on the other atoms nearby and their locations. Consider for instance solid sodium chloride- table salt. The sodium chloride crystal has atoms arranged on a cubic lattice, but with alternating sodium and chlorine atoms, the sodium ones having a single positive charge \(+e\) and the chlorine ones a single negative charge \(-e\), where \(e\) is the charge on the electron. If we label each position on the lattice by three integer coordinates \((i, j, k)\), then the sodium atoms fall at positions where \(i+j+k\) is even, and the chlorine atoms at positions where \(i+j+k\) is odd. Consider a sodium atom at the origin, \(i=j=k=0\), and let us calculate the Madelung constant. If the spacing of atoms on the lattice is \(a\), then the distance from the origin to the atom at position \((i, j, k)\) is $$ \sqrt{(i a)^{2}+(j a)^{2}+(k a)^{2}}=a \sqrt{i^{2}+j^{2}+k^{2}} $$ and the potential at the origin created by such an atom is $$ V(i, j, k)=\pm \frac{e}{4 \pi e_{0} a \sqrt{i^{2}+j^{2}+k^{2}}}, $$ with \(e_{0}\) being the permittivity of the vacuum and the sign of the expression depending on whether \(i+j+k\) is even or odd. The total potential felt by the sodium atom is then the sum of this quantity over all other atoms. Let us assume a cubic box around the sodium at the origin, with \(L\) atoms in all directions. Then $$ V_{\text {beal }}=\sum_{n, j, i=-t \atop n=0}^{L} V(i, j, k)=\frac{e}{4 \pi e_{0} a} M \text {, } $$ where \(M\) is the Madelung constant, at least approximately-technically the Madelung constant is the value of \(M\) when \(L \rightarrow \infty\), but one can get a good approximation just by using a large value of \(L\). Write a program to calculate and print the Madelung constant for sodium chloride. Use as large a value of \(L\) as you can, while still having your program run in reasonable time-say in a minute or less.

Altitude of a satellite A satellite is to be launched into a circular orbit around the Earth so that it orbits the planet once every \(T\) seconds. a) Show that the altitude \(h\) above the Earth's surface that the satellite must have is $$ h=\left(\frac{G M T^{2}}{4 \pi^{2}}\right)^{1 / 3}-R $$ where \(G=6.67 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) is Newton's gravitational constant, \(M=\) \(5.97 \times 10^{24} \mathrm{~kg}\) is the mass of the Earth, and \(R=6371 \mathrm{~km}\) is its radius. b) Write a program that asks the user to enter the desired value of \(T\) and then calculates and prints out the correct altitude in meters. c) Use your program to calculate the altitudes of satellites that orbit the Earth once a day (so-called "geosynchronous" orbit), once every 90 minutes, and once every 45 minutes. What do you conclude from the last of these calculations? d) Technically a geosynchronous satellite is one that orbits the Earth once per sidereal day, which is \(23.93\) hours, not 24 hours. Why is this? And how much difference will it make to the altitude of the satellite?

Catalan numbers The Catalan numbers \(C_{n}\) are a sequence of integers \(1,1,2,5,14,42,132 \ldots\) that play an important role in quantum mechanics and the theory of disordered systems. (They were central to Eugene Wigner's proof of the so- called semicircle law.) They are given by $$ C_{0}=1, \quad C_{n+1}=\frac{4 n+2}{n+2} C_{n} . $$ Write a program that prints in increasing order all Catalan numbers less than or equal to one billion.

Planetary orbits The orbit in space of one body around another, such as a planet around the Sun, need not be circular. In general it takes the form of an ellipse, with the body sometimes closer in and sometimes further out. If you are given the distance \(\ell_{1}\) of closest approach that a planet makes to the Sun, also called its perihelion, and its linear velocity \(v_{1}\) at perihelion, then any other property of the orbit can be calculated from these two as follows. a) Kepler's second law tells us that the distance \(\ell_{2}\) and velocity \(v_{2}\) of the planet at its most distant point, or aphelion, satisfy \(\ell_{2} v_{2}=\ell_{1} v_{1}\). At the same time the total energy, kinetic plus gravitational, of a planet with velocity \(v\) and distance \(r\) from the Sun is given by $$ E=\frac{1}{2} m v^{2}-G \frac{m M}{r}, $$ where \(m\) is the planet's mass, \(M=1.9891 \times 10^{30} \mathrm{~kg}\) is the mass of the Sun, and \(G=6.6738 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) is Newton's gravitational constant. Given that energy must be conserved, show that \(v_{2}\) is the smaller root of the quadratic equation $$ v_{2}^{2}-\frac{2 G M}{v_{1} \ell_{1}} v_{2}-\left[v_{1}^{2}-\frac{2 G M}{\ell_{1}}\right]=0 . $$ Once we have \(v_{2}\) we can calculate \(\ell_{2}\) using the relation \(\ell_{2}=\ell_{1} v_{1} / v_{2}\). b) Given the values of \(v_{1}, \ell_{1}\), and \(\ell_{2}\), other parameters of the orbit are given by simple formulas can that be derived from Kepler's laws and the fact that the orbit is an ellipse: $$ \begin{aligned} \text { Semi-major axis: } & a=\frac{1}{2}\left(\ell_{1}+\ell_{2}\right), \\ \text { Semi-minor axis: } & b=\sqrt{\ell_{1} \ell_{2}}, \\ \text { Orbital period: } & T=\frac{2 \pi a b}{\ell_{1} v_{1}}, \\ \text { Orbital eccentricity: } & e=\frac{\ell_{2}-\ell_{1}}{\ell_{2}+\ell_{1}} . \end{aligned} $$ Write a program that asks the user to enter the distance to the Sun and velocity at perihelion, then calculates and prints the quantities \(\ell_{2}, v_{2}, T\), and \(e\). c) Test your program by having it calculate the properties of the orbits of the Earth (for which \(\ell_{1}=1.4710 \times 10^{11} \mathrm{~m}\) and \(v_{1}=3.0287 \times 10^{4} \mathrm{~ms}^{-1}\) ) and Halley's comet \(\left(\ell_{1}=8.7830 \times 10^{10} \mathrm{~m}\right.\) and \(\left.v_{1}=5.4529 \times 10^{4} \mathrm{~m} \mathrm{~s}^{-1}\right)\). Among other things, you should find that the orbital period of the Earth is one year and that of Halley's comet is about 76 years.
See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Atoms and Radioactivity

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Wave Optics

Read Explanation

Engineering Physics

Read Explanation

Physics of Motion

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free