Abstract
This chapter introduces the basics of the techniques for the study of the intimate structure of matter, described in a historical context. After reading this chapter, you should understand the basic tools which lead to the investigation and the description of the subatomic structure, and you should be able to compute the interaction probabilities of particles. A short reminder of the concepts of special relativity needed to understand astroparticle physics is also provided.
This chapter introduces the basics of the techniques for the study of the intimate structure of matter, described in a historical context. After reading this chapter, you should understand the basic tools which lead to the investigation and the description of the subatomic structure, and you should be able to compute the interaction probabilities of particles. A short reminder of the concepts of special relativity needed to understand astroparticle physics is also provided.
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Notes
- 1.
Dimitri Mendeleev (1834–1907) was a Russian chemist born in Tobolsk, Siberia. He studied science in St. Petersburg, where he graduated in 1856 and became full professor in 1863. Mendeleev is best known for his work on the periodic table, published in Principles of Chemistry in 1869, but also, according to a myth popular in Russia, for establishing that the minimum alcoholic fraction of vodka should be 40 %—this requirement was easy to verify, as this is the minimum content at which an alcoholic solution can be ignited at room temperature.
- 2.
Ernest Rutherford (1871–1937) was a New Zealand-born physicist. In early works at McGill University in Canada, he proved that radioactivity involved the transmutation of one chemical element into another; he differentiated and named the \(\alpha \) (helium nuclei) and \(\beta \) (electrons) radiations. In 1907, Rutherford moved to Manchester, UK, where he discovered (and named) the proton. In 1908 he won the Nobel Prize in Chemistry “for his investigations into the disintegration of the elements, and the chemistry of radioactive substances.” He became director of the Cavendish Laboratory at Cambridge University in 1919. Under his leadership, the neutron was discovered by James Chadwick in 1932. Also in 1932, his students John Cockcroft and Ernest Walton split for the first time the atom with a beam of particles. Rutherford was buried near Newton in Westminster Abbey, London. The chemical element rutherfordium—atomic number 104—was named after him in 1997.
- 3.
Enrico Fermi (Rome 1901–Chicago 1954) studied in Pisa and became full professor of Analytical Mechanics in Florence in 1925, and then of Theoretical Physics in Rome from 1926. Soon he surrounded himself by a group of brilliant young collaborators, the so-called “via Panisperna boys” (E. Amaldi, E. Majorana, B. Pontecorvo, F. Rasetti, E. Segré, O. D’Agostino). For Fermi, theory and experiment were inseparable. In 1934, he discovered that slow neutrons catalyzed a certain type of nuclear reactions, which made it possible to derive energy from nuclear fission. In 1938, Fermi went to Stockholm to receive the Nobel Prize, awarded for his fundamental work on neutrons, and from there he emigrated to the USA, where he became American citizen in open dispute with the Italian racial laws. He actively participated in the Manhattan Project for the use of nuclear power for the atomic bomb, but spoke out against the use of this weapon on civilian targets. Immediately after the end of World War II, he devoted himself to theoretical physics of elementary particles and to the origin of cosmic rays. Few scientists of the twentieth century impacted as profoundly as Fermi in different areas of physics: Fermi stands for elegance and power of thought in the group of immortal geniuses like Einstein, Landau, Heisenberg, and later Feynman.
- 4.
Richard Feynman (New York 1918—Los Angeles 1988), longtime professor at Caltech, is known for his work in quantum mechanics, in the theory of quantum electrodynamics, as well as in particle physics; he participated in the Manhattan project. In addition, he proposed quantum computing. He received the Nobel Prize in Physics in 1965 for his “fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles.” His life was quite adventurous, and full of anecdotes. In the divorce file related to his second marriage, his wife complained that “He begins working calculus problems in his head as soon as he awakens. He did calculus while driving in his car, while sitting in the living room, and while lying in bed at night.” He wrote several popular physics books, and an excellent general physics textbook now freely available at http://www.feynmanlectures.caltech.edu/.
- 5.
Hideki Yukawa (Tokyo, 1907–Kyoto, 1981) , professor at the Kyoto University, gave fundamental contributions to quantum mechanics. For his research he won the prize Nobel Prize for Physics in 1949.
- 6.
James Clerk Maxwell (1831–1879) was a Scottish physicist. His most prominent achievement was formulating classical electromagnetic theory. Maxwell’s equations, published in 1865, demonstrate that electricity, magnetism, and light are all manifestations of the same phenomenon: the electromagnetic field. Maxwell also contributed to the Maxwell-Boltzmann distribution, which gives the statistical distribution of velocities in a classical perfect gas in equilibrium. Einstein had a photograph of Maxwell, one of Faraday and one of Newton in his office.
- 7.
Hendrik Antoon Lorentz (1853–1928) was a Dutch physicist who gave important contributions to electromagnetism. He also derived the equations subsequently used by Albert Einstein to describe the transformation of space and time coordinates in different inertial reference frames. He was awarded the 1902 Nobel Prize in Physics.
- 8.
Ludvig Lorenz (1829–1891) , not to be confused with Hendrik Antoon Lorentz, was a Danish mathematician and physicist, professor at the Military Academy in Copenhagen.
- 9.
For reasons related only to metrology (reproducibility and accuracy of the definition) in the standard SI the unit of electrical current, the ampere A, is used instead of the coulomb; the two definitions are however conceptually equivalent.
- 10.
\(\hbar c\simeq 1.97\times 10^{-13}\mathrm{{MeV \cdot m}=3.15\times 10^{-26}\mathrm{{J \cdot m.}}}\)
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Appendices
Further Reading
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[F2.1] J.S. Townsend, “A modern approach to quantum mechanics”, Mc Graw-Hill 2012. An excellent quantum mechanics course at an advanced undergraduate level.
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[F2.2] W. Rindler, “Introduction to Special Relativity”, second edition, Oxford University Press 1991. A classical textbook on special relativity for undergraduates.
Exercises
-
1.
Rutherford formula. Consider the Rutherford formula.
-
(a)
Determine the distance of closest approach of an \(\alpha \) particle with an energy of 7.7 MeV to a gold target .
-
(b)
Determine the de Broglie wavelength of that \(\alpha \) particle.
-
(c)
Explain why the classical Rutherford formula survived the revolution of quantum mechanics.
You can find the numerical values of particle data and fundamental constants in the Appendices, or in your Particle Data Book(let).
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(a)
-
2.
Cross-section at fixed target. Consider a fixed target experiment with a monochromatic proton beam with an energy of 20 GeV and a 2 m long liquid hydrogen \((H_2)\) target (\(\rho = 60\) kg/m\(^3\)). In the detector placed just behind the target beam fluxes of \(7 \times 10^6\) protons/s and \(10^7\) protons/s are measured, respectively, with the target full and empty. Determine the proton–proton total cross-section at this energy and its statistical error:
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(a)
without taking into account the attenuation of the beam inside the target;
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(b)
taking into account the attenuation of the beam inside the target.
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(a)
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3.
LHC collisions. The LHC running parameters in 2012 were, for a c.m. energy \(\sqrt{s} \simeq \) 8 TeV: number of bunches = 1400; time interval between bunches \( \simeq 50\) ns; number protons per bunch \(\simeq \) 1.1 \(\times 10^{11}\); beam width at the crossing point \( \simeq 16\,\upmu \)m.
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(a)
Determine the maximum instantaneous luminosity of the LHC in 2012.
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(b)
Determine the number of interactions per collision (\(\sigma _{pp} \sim 100\) mb).
-
(c)
As you probably heard, LHC found a particle called Higgs boson, which Leon Lederman called the “God particle” (a name the news like very much). If Higgs bosons are produced with a cross-section \(\sigma _H \sim 21\) pb, determine the number of Higgs bosons decaying into 2 photons (\(BR(H\rightarrow \gamma \gamma ) \simeq 2.28 \times 10^{-3}\)) which might have been produced in 2012 in the LHC, knowing that the integrated luminosity of the LHC (luminosity integrated over the time) during 2012 was around 20 fb\(^{-1}\). Compare it to the real number of detected Higgs in this particular decay mode reported by the LHC collaborations (about 400). Discuss the difference.
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(a)
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4.
Classical electromagnetism is not a consistent theory. Consider two electrons at rest, and let r be the distance between them. The (repulsive) force between the two electrons is the electrostatic force
$$\begin{aligned} F =\frac{1}{4\pi \epsilon _0}\frac{e^2}{r^2} \, , \end{aligned}$$where e is the charge of the electron; it is directed along the line joining the two charges. But an observer is moving with a velocity v perpendicular to the line joining the two charges will measure also a magnetic force (still directed as F)
$$\begin{aligned} F'=\frac{1}{4\pi \epsilon _0}\frac{e^2}{r^2}-\frac{\mu _0}{2\pi r}v^2e^2 \ne F \, . \end{aligned}$$The expression of the force is thus different in the two frames of reference. But masses, charges, and accelerations are classically invariant. Comment.
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5.
GZK threshold. The Cosmic Microwave Background fills the Universe with photons with a peak energy of 0.37 meV and a density of \(\rho \sim \) 400/cm\(^3\). Determine:
-
(a)
The minimal energy (known as the GZK threshold) that a proton should have in order that the reaction \(p \gamma \rightarrow \Delta \) may occur.
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(b)
The interaction length of such protons in the Universe considering a mean cross-section above the threshold of 0.6 mb.
-
(a)
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6.
\(\bar{p}\) production at the Bevatron. The antiprotons were first produced in the laboratory in 1955, in proton–proton fixed target collisions at an accelerator called Bevatron (it was named for its ability to impart energies of billions of eV, i.e., Billions of eV Synchrotron), located at Lawrence Berkeley National Laboratory, US . The discovery resulted in the 1959 Nobel Prize in physics for Emilio Segrè Owen Chamberlain .
-
(a)
Describe the minimal reaction able to produce antiprotons in such collisions.
-
(b)
When a proton is confined in a nucleus, it cannot have arbitrarily low momenta, as one can understand from the Heisenberg principle; the actual value of its momentum is called the “Fermi momentum.” Determine the minimal energy that the proton beam must have in order that antiprotons were produced considering that the target protons have a Fermi momentum of around 150 MeV/c.
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(a)
-
7.
Photon conversion. Consider the conversion of one photon in one electron–positron pair. Determine the minimal energy that the photon has to have in order that this conversion would be possible if the photon is in presence of:
-
(a)
one proton;
-
(b)
one electron;
-
(c)
when no charged particle is around.
-
(a)
-
8.
\(\pi ^-\) decay. Consider the decay of a flying \(\pi ^-\) into \(\mu ^- \bar{\nu _\mu }\) and suppose that the \(\mu ^-\) was emitted along the flight line of flight of the \(\pi ^-\). Determine:
-
(a)
The energy and momentum of the \(\mu ^-\) and of the \(\bar{\nu _\mu }\) in the \(\pi ^-\) frame.
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(b)
The energy and momentum of the \(\mu ^-\) and of the \(\bar{\nu _\mu }\) in the laboratory frame, if the momentum \(P_\pi ^-\) = 100 GeV/c.
-
(c)
Same as the previous question but considering now that was the \(\bar{\nu _\mu }\) that was emitted along the flight line of the \(\pi ^-\).
-
(a)
-
9.
\(\pi ^0\) decay. Consider the decay of a \(\pi ^0\) into \(\gamma \gamma \) (with pion momentum of 100 GeV/c). Determine:
-
(a)
The minimal and the maximal angles between the two photons in the laboratory frame.
-
(b)
The probability of having one of the photons with an energy smaller than an arbitrary value \(E_0\) in the laboratory frame.
-
(c)
Same as (a) but considering now that the decay of the \(\pi ^0\) is into \(e^+e^-\).
-
(d)
The maximum momentum that the \(\pi ^0\) may have in order that the maximal angle in its decay into \(\gamma \gamma \) and in \(e^+e^-\) would be the same.
-
(a)
-
10.
Invariant flux. In a collision between two particles a and b the incident flux is given by \(F = 4|\vec {v_a}-\vec {v_b}| E_a E_b\) where \(\vec {v_a}\), \(\vec {v_b}\), \(E_a\) and \(E_b\) are, respectively, the vectorial speeds and the energies of particles a and b.
-
(a)
Verify that the above formula is equivalent to: \(F = 4 \sqrt{(P_a P_b)^2 - (m_a m_b)^2 }\) where \(P_a\) and \(P_b\) are, respectively, the four-vectors of particles a and b, and \(m_a\) and \(m_b\) their masses.
-
(b)
Relate the expressions of the flux in the center-of-mass and in the laboratory reference frames.
-
(a)
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11.
Units. Determine in Natural Units:
-
(a)
Your own dimensions (height, weight, mass, age).
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(b)
The mean lifetime of the muon (\(\tau _\mu =2.2 \, \mu s\)).
-
(a)
-
12.
Units. In NU the expression of the muon life time is given as
$$ \tau _\mu = \frac{192 \pi ^3}{G_F^2 m_\mu ^5} $$where \(G_F\) is the Fermi constant.
-
(a)
Is the Fermi constant dimensionless? If not compute its dimension in NU and in SI.
-
(b)
Obtain the conversion factor for transforming \(G_F\) from SI to NU.
-
(a)
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De Angelis, A., Pimenta, M.J.M. (2015). The Birth and the Basics of Particle Physics. In: Introduction to Particle and Astroparticle Physics. Undergraduate Lecture Notes in Physics. Springer, Milano. https://doi.org/10.1007/978-88-470-2688-9_2
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