Mathematical Physics for Nuclear Experiments

Andrew E. Ekpenyong

• Publisher: CRC Press
• Year: 2022
• ISBN:
  9781000464061
• Edition: 1
• Topic: Ciencias físicas
• Subtopic: Física nuclear

Mathematical Physics for Nuclear Experiments presents an accessible introduction to the mathematical derivations of key equations used in describing and analysing results of typical nuclear physics experiments. Instead of merely showing results and citing texts, crucial equations in nuclear physics such as the Bohr's classical formula, Bethe's quantum mechanical formula for energy loss, Poisson, Gaussian and Maxwellian distributions for radioactive decay, and the Fermi function for beta spectrum analysis, among many more, are presented with the mathematical bases of their derivation and with their physical utility.

This approach provides readers with a greater connection between the theoretical and experimental sides of nuclear physics. The book also presents connections between well-established results and ongoing research. It also contains figures and tables showing results from the author's experiments and those of his students to demonstrate experimental outcomes.

This is a valuable guide for advanced undergraduates and early graduates studying nuclear instruments and methods, medical and health physics courses as well as experimental particle physics courses.

Key features

• Contains over 500 equations connecting theory with experiments.
• Presents over 80 examples showing physical intuition and illustrating concepts.
• Includes 80 exercises, with solutions, showing applications in nuclear and medical physics.

CHAPTER 1 Radioactivity and Decay Law

DOI: 10.1201/9781003215622-1

Alan Turing's words that science is a differential equation seem quite true in the physics of radioactivity. This chapter begins with a differential equation called the radioactive decay law and ends with equations for specific decay processes in the unstable nucleus including alpha decay, beta decay and gamma decay. Throughout the chapter, we give mathematical derivations, equations and example calculations which enable us to make sense of (theory) and make use of (applications) the results of nuclear experiments.

1.1THE RADIOACTIVE DECAY LAW

The radioactive decay law simply states that the activity of a radioactive sample decays exponentially in time. This law implies that the rate at which a radioactive sample decays (activity) is proportional to the remaining number of atoms. Derived experimentally by Ernest Rutherford (1871–1937) and Frederick Soddy (1877–1956), it can also be derived from Quantum Mechanics in terms of a transition probability per unit time, λ, which is characteristic of the nuclear species. Since λ is a constant, independent of time, it is aptly termed decay constant. Thus, in a sample of $N(t)$ radioactive nuclei, Eq. 1.1a gives the mean number of nuclei dN, decaying in a time dt:

$$\begin{array}{l}dN=-\lambda Ndt.\hfill \end{array}$$(1.1a)

$$\begin{array}{l}\frac{dN}{dt}=-\lambda N.\hfill \end{array}$$(1.1b)

$$\begin{array}{l}\frac{dN}{N}=-\lambda dt.\hfill \end{array}$$(1.1c)

Eq. 1.1b states the decay law as a differential equation, making it obvious as a separable equation which can therefore be written in the form given by Eq. 1.1c. Integrating both sides of Eq. 1.1c, we g...