Chapter 1 — Summary, Toolkit and Problems · Elementary Particle Physics

The chapter gave two ways to answer any kinematics question — boost everything, or compute an invariant. Thirty-eight problems are where the second stops being a preference and becomes the only practical route.

🎯 Why this matters

Three printed results in this chapter do not survive being redone. That is not a complaint about the book; it is why the solutions here are worked rather than quoted, and why any number you intend to rely on is worth recomputing.

Chapter 1 was the toolbox. This page is where you find out whether you can use it: the book’s own summary, the two-body kinematics formulary it gives you before the problems, and all 38 problems with worked solutions.

What you were supposed to take away

The book’s summary, expanded into what each item actually means and where it was built:

The chapter's summary (p. 69), with the section that owns each item

You should now have↕	What that means in practice↕	§
the Lorentz transformations from general properties of space-time	Derived from five structural axioms — homogeneity, isotropy, relativity, the group property, causality — with light entering only at the end to fix c. The point is that this constrains ALL interactions, not just electromagnetism.	1.1
mass, energy and momentum, their transformation properties, and the invariants	m² = E² − p² defines mass as the norm of a 4-vector; "relativistic mass" does not exist; the mass of a system is not the sum of its parts.	1.2–1.5
the L and CM frames	Fixed target versus collider, and the quadratic penalty that decides which machine gets built.	1.4–1.5, 1.12
SI and natural units	ħ = c = 1, and the ability to put the factors back. ħc = 197.3 MeV fm is the constant you will use most.	1.6–1.7
cross-section, luminosity, decay rates, branching ratios, phase space	σ is an effective area; Lσ is a rate; Γτ = ħ; the golden rule splits any rate into a matrix element and a phase-space volume.	1.6–1.7
the basic aspects of a scattering experiment	Scattering is a Fourier transform: resolution costs momentum transfer, which costs beam energy. Form factors, Rutherford, Mott.	1.8
the names of the particle types and of the four interactions	The census: quarks, leptons, gauge bosons, hadrons; and why gravity never appears again.	1.9–1.10
how charged particles and photons lose energy and are detected	Bethe–Bloch and its 1/β² rise, bremsstrahlung and X₀, critical energy, λ₀, showers.	1.11
the sources: cosmic rays, accelerators, colliders	p = 0.3BR sizes every machine ever built; phase stability and stochastic cooling are what make them work.	1.12
the basic detector types, tracking and calorimetry	Eleven instruments, each exploiting one of the energy-loss mechanisms above — and each dying, eventually, of rate.	1.13a–d

If any row reads as unfamiliar, the problems below will find it — the topic filter maps onto these rows.

The two-body toolkit

Before the problems the book works out, once and for all, every kinematic quantity of a generic two-body scattering $a + b \to c + d$ in terms of the invariants. These are equations (P1.1)–(P1.15), and they are worth having at hand rather than re-deriving: most of the 38 problems are one substitution into one of them.

Everything rests on two facts from §1.4–1.5: $s$ , $t$ and $u$ are the same in every frame, and each of them can be written either from the initial state or from the final state.

Before the algebra, the picture the formulary is about — and the labelling the book uses without ever drawing it:

Draw this before starting any problem. The book’s solution to Problem 1.15 opens by saying so, and it is the single most useful habit in the chapter: nearly every sign error in relativistic kinematics is a mislabelled angle.

\htmlClass{t-s}{s} = \left(\htmlClass{t-Ea}{E_a}+\htmlClass{t-mb}{m_b}\right)^2 - \htmlClass{t-pa}{p_a}^2 = \htmlClass{t-ma}{m_a}^2 + \htmlClass{t-mb}{m_b}^2 + 2\,\htmlClass{t-Ea}{E_a}\htmlClass{t-mb}{m_b} \quad\Longrightarrow\quad \htmlClass{t-Ea}{E_a} = \frac{\htmlClass{t-s}{s} - \htmlClass{t-ma}{m_a}^2 - \htmlClass{t-mb}{m_b}^2}{2\,\htmlClass{t-mb}{m_b}}

(P1.1)

Read left to right for the CM energy of a fixed-target machine; read the arrow for the beam energy you need to reach a given s. Both directions get used constantly.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

The formulary — (P1.2) to (P1.15), for a + b → c + d

Eq.↕	Result↕	What it is for↕
P1.2–P1.3	E_b = (s + m_b² − m_a²)/2√s ; E_a = (s + m_a² − m_b²)/2√s	CM energies of the two incoming particles. Note they sum to √s, as they must.
P1.4	p_a = p_b = √(E*² − m²)	The common CM momentum of the initial state — equal and opposite by definition of the frame.
P1.5–P1.7	E_c = (s + m_c² − m_d²)/2√s ; E_d = (s + m_d² − m_c²)/2√s ; p_c = p_d
P1.8	t = m_c² + m_a² + 2p_a p_c cos θ_ac − 2E_a E_c = m_d² + m_b² + 2p_b p_d cos θ_bd − 2E_b E_d	The momentum transfer written two ways — through the a→c vertex or the b→d vertex. Both are the same invariant.
P1.9–P1.10	cos θ_ac = (t − m_a² − m_c² + 2E_a E_c)/(2p_a p*_c)	The scattering angle, extracted from t. See the warning below about how P1.9 is printed.
P1.11–P1.12	in L (p_b = 0): t = m_b² + m_d² − 2m_b E_d ⇒ E_d = (m_b² + m_d² − t)/2m_b	The recoil energy of the target from the momentum transfer alone. This is the elastic-scattering workhorse of §1.8.
P1.13	E_c = m_b + E_a − E_d = (s + t − m_a² − m_d²)/2m_b	The scattered particle's lab energy. Used in Problem 1.17 to get the opening angle without a single Lorentz transformation.
P1.14–P1.15	u = m_d² + m_a² + 2p_a p_d cos θ_ad − 2E_a E_d ; in L, u = m_b² + m_c² − 2m_b E_c	The third invariant, and its simple lab form. With P1.13 it gives s + t + u = Σm² immediately.

Six of these are the same formula with the labels permuted — which is the point. <strong>Learn the pattern, not the fourteen equations.</strong> Every energy is (s ± m² ∓ m²)/2√s in the CM, or (something − t)/2m_b in the lab, and every momentum is √(E² − m²).

⚠️ Equation (P1.9) as printed

The book prints

\cos\theta^*_{ac} = \frac{t - m_a^2 - m_c^2 + 2\,m_a m_c}{2\,p^*_a p^*_c},

with masses in the last term. Its neighbour (P1.10) has energies in the same slot, and so does the equation it is derived from. Solving (P1.8) for the cosine gives

2 p_a p_c \cos\theta_{ac} = t - m_a^2 - m_c^2 + 2E_aE_c \quad\Longrightarrow\quad \cos\theta^*_{ac} = \frac{t - m_a^2 - m_c^2 + 2E^*_aE^*_c}{2\,p^*_a p^*_c},

so the printed $2m_am_c$ should be $2E^*_aE^*_c$ . The two coincide only when both particles are at rest, which is exactly the case where the angle is undefined.

Nothing later depends on it — Problem 1.17’s solution uses the correct relation in the equivalent form $t = 2m_p^2 + 2p^{*2}\cos\theta^* - 2E^{*2}$ — but the formula as printed will not reproduce that problem’s answer.

⚙️ Engineer’s bridge — why invariants beat transformations

Problem 1.17 is deliberately solved twice: once by boosting every 4-vector into the CM and back, and once by computing $s$ and $t$ and substituting. The second route is shorter, and the reason will be familiar.

A Lorentz transformation is a change of basis. Carrying a problem through one means tracking every component of every vector through a matrix multiply, and every component is an opportunity to drop a sign. The invariants are basis-independent scalars — the checksums of the problem. You compute them once, in whichever frame is easiest, and they are then valid everywhere.

It is the same instinct as working with a norm, a determinant or a trace instead of the matrix entries: if the answer you want is a scalar, find a scalar route to it. And when you do need both routes, they are a free cross-check — which is precisely how the book uses them here.

Where it breaks: an invariant route exists only when the answer is a scalar, and a great deal of what this book measures is not. Anything angular — the $1+\cos^2\theta$ distribution of §5.7, a Dalitz plot, a forward–backward asymmetry — is frame-dependent by construction, so you must name the frame and do the transformation. The rule is therefore narrower than it sounds: prefer invariants for quantities, and expect to work in a specified frame the moment the question is about a direction.

Reproduce it

import numpy as np

mp = 0.93827209                                   # GeV
p1, th = 3.0, np.radians(10.0)                    # Problem 1.17
E1 = np.hypot(p1, mp)
s  = (E1 + mp)**2 - p1**2
print(f"P1.1 check: s = {s:.4f} GeV^2 from (E_a+m_b)^2-p_a^2, "
      f"and E_a back out = {(s - 2*mp**2)/(2*mp):.4f} GeV")

rs   = np.sqrt(s)
Ecm  = rs/2                                       # all four particles equal mass
pcm  = np.sqrt(Ecm**2 - mp**2)
t    = 2*pcm**2*(np.cos(th) - 1)                  # elastic, equal masses

bad  = (t - 2*mp**2 + 2*mp*mp)     /(2*pcm*pcm)   # (P1.9) as printed
good = (t - 2*mp**2 + 2*Ecm*Ecm)   /(2*pcm*pcm)   # with energies
print(f"P1.9 as printed (masses)  : cos theta* = {bad:+.4f} -> {np.degrees(np.arccos(bad)):.1f} deg")
print(f"P1.9 with energies (right) : cos theta* = {good:+.4f} -> "
      f"{np.degrees(np.arccos(good)):.1f} deg   <- the input angle")

u = 4*mp**2 - s - t                               # from the identity, then verify
print(f"s + t + u = {s+t+u:.4f} GeV^2 ;  sum of m^2 = {4*mp**2:.4f} GeV^2"
      f"   -> identity holds")

prints

P1.1 check: s = 7.6593 GeV^2 from (E_a+m_b)^2-p_a^2, and E_a back out = 3.1433 GeV
P1.9 as printed (masses)  : cos theta* = -0.0152 -> 90.9 deg
P1.9 with energies (right) : cos theta* = +0.9848 -> 10.0 deg   <- the input angle
s + t + u = 3.5214 GeV^2 ;  sum of m^2 = 3.5214 GeV^2   -> identity holds

$s+t+u=\sum m^2$ is not a formula to memorise, it is a constraint: it collapses three invariants to two. Fix $s$ and every possible event of this reaction lies somewhere on one straight line; the requirement that both $t$ and $u$ be negative cuts that line down to the coloured segment, whose two endpoints are forward and backward scattering. The elastic peak everyone measures sits at the crowded right-hand end — Problem 1.17’s 10° is already almost on top of the forward endpoint.

The problems

All 38, with a hint before the solution. The topic chips filter them; the progress bar and the “solved” ticks are stored in your browser.

📝 Chapter 1 problems

0/38 solved

1.1unitstheory
Estimate the kinetic energy of a Boeing 747 at cruising speed and compare it with the energy released by a mosquito annihilating with an antimosquito.
- $M = 400$ t
- $v = 850$ km/h
- mosquito mass $\approx 2.5$ mg (assumed — the book leaves it to you)
1.2kinematicstheory
Three protons have momenta equal in magnitude and directions at $120°$ from one another. What is the mass of the system?
- $|\mathbf p| = 3$ GeV each
1.3unitstheory
Compute the widths of the weak decays of the $\pi^\pm$ , $K^\pm$ and $\Lambda$ .
- $\tau_\pi = 26$ ns
- $\tau_K = 12$ ns
- $\tau_\Lambda = 0.26$ ns
1.4unitstheory
Compute the lifetimes of the strongly decaying $\rho$ , $\omega$ , $\phi$ , $K^*$ , $J/\psi$ and $\Delta$ from their total widths.
- $\Gamma_\rho = 149$ MeV
- $\Gamma_\omega = 8.5$ MeV
- $\Gamma_\phi = 4.3$ MeV
- $\Gamma_{K^*} = 51$ MeV
- $\Gamma_{J/\psi} = 93$ keV
- $\Gamma_\Delta = 118$ MeV
1.5scatteringtheory
An accelerator produces a 20 GeV electron beam. Electrons scattered at $\theta = 6°$ are detected. Neglecting the recoil, what is the smallest structure in the proton that can be resolved?
- $E = 20$ GeV
- $\theta = 6°$
1.6kinematicstheory
In a $pp$ collision the final state contains a particle of mass $m$ besides the two protons. (a) Give the threshold energy $E_p$ and momentum $p_p$ for a proton target at rest. (b) Give $E_p^*$ and $p_p^*$ for two protons colliding with equal and opposite velocities. (c) Evaluate both for a produced pion, and give the kinetic energy in case (a).
1.7kinematicstheory
Consider $\gamma + p \to p + \pi^0$ on a proton at rest. (a) Find the threshold photon energy. (b) The Universe is filled with 3 K background radiation of photon energy $E_\gamma \approx 1$ meV; find the minimum energy a cosmic-ray proton needs to induce $\pi^0$ photoproduction on it. (c) With $\sigma = 0.6$ mb just above threshold and a background photon density $\rho \approx 10^8\ \text{m}^{-3}$ , find the attenuation length.
1.8kinematicstheory
The Universe is opaque to photons energetic enough that $\gamma\gamma \to e^+e^-$ can occur on a background photon. Compute the threshold energy against (a) the 3 K microwave background at $E_\gamma \approx 1$ meV and (b) the extragalactic background light, taking $\lambda = 1\ \mu$ m.
1.9kinematicstheory
The Bevatron was designed to have enough energy to produce antiprotons. What is the minimum proton beam energy? Baryon number conservation forces the reaction $p + p \to p + p + \bar p + p$ .
1.10kinematicstheory
At the LHC two proton beams of $E_p = 7$ TeV collide head on. What beam energy would give the same centre-of-mass energy on a fixed hydrogen target? How does it compare with cosmic-ray energies?
1.11kinematicstheory
A particle of mass $M$ decays to two bodies of masses $m_1$ and $m_2$ . Give the energies and momenta of the products in the CM frame.
1.12kinematicstheory
Evaluate the CM energies and momenta of the products of $\Lambda \to p\pi^-$ and $\Xi^- \to \Lambda\pi^-$ .
1.13kinematicstheory
Find the energies and momenta of the products of $M \to m_1 + m_2$ in the CM when $m_2 = 0$ .
1.14kinematicstheory
In a monochromatic $\pi$ beam of momentum $p_\pi$ some pions decay in flight as $\pi \to \mu\nu_\mu$ . We observe that in some cases the muons move backwards. Find the maximum $p_\pi$ for which this is possible.
1.15kinematicstheory
A $\Lambda$ of momentum $p_\Lambda = 2$ GeV decays as $\Lambda \to p\pi^-$ . In the CM the proton makes an angle $\theta^*_p = 30°$ with the $\Lambda$ direction. Find (a) the CM energies and momenta, (b) the Lorentz parameters of the L–CM transformation, (c) the lab energy and momentum of the $\pi$ , and the lab angle and momentum of the $p$ .
1.16kinematicstheory
A ball collides elastically with an equal ball at rest. Compute the angle between the two final directions at non-relativistic speeds.
1.17kinematicstheory
A proton of momentum $p_1 = 3$ GeV scatters elastically on a proton at rest; one proton comes out at $\theta^*_{ac} = 10°$ in the CM. Find (a) the kinematic quantities in the L frame, (b) those in the CM, (c) the angle between the final protons in L — is it $90°$ ?
1.18detectorstheory
A charmed meson $D^0$ decays as $D^0 \to K^-\pi^+$ at $d = 3$ mm from its production point. The total energy of the decay products is measured as $E = 30$ GeV. How long did the $D$ live in proper time, and what is the $\pi^+$ momentum in the $D$ rest frame?
- $d = 3$ mm
- $E = 30$ GeV
- $m_D = 1.865$ GeV
1.19detectorstheory
A secondary monochromatic $\pi^-$ beam is produced at a target. At $l = 20$ m from the target, 10 % of the pions have decayed. Find the momentum and energy of the pions.
- $l = 20$ m
- 10 % decayed
- $c\tau_\pi = 7.80$ m
1.20kinematicstheory
A $\pi^-$ beam is stopped in liquid hydrogen, where $\pi^0$ are produced by charge exchange, $\pi^- + p \to \pi^0 + n$ . Find the energy of the $\pi^0$ , the kinetic energy of the neutron, the velocity of the $\pi^0$ and the distance it travels in one lifetime.
1.21scatteringtheory
A 2 GeV electron beam hits an iron target (take pure $^{56}$ Fe). How large is the maximum four-momentum transfer?
- $E = 2$ GeV
- $M \approx 56$ GeV
1.22scatteringtheory
Geiger and Marsden found that alpha particles bounced backwards off a thin foil 'not too infrequently'. Calculate the ratio between the scattering probabilities for $\theta > 90°$ and for $\theta > 10°$ .
1.23scatteringtheory
A 6 MeV alpha beam of intensity $R_i = 10^3\ \text{s}^{-1}$ crosses a $1\ \mu$ m gold foil. Calculate the number of particles per unit time scattered at angles larger than 0.1 rad.
- $Z = 79$ , $A = 197$
- $\rho = 1.93\times10^4\ \text{kg m}^{-3}$
- $t = 1\ \mu$ m
1.24scatteringtheory
Electrons of 10 GeV are scattered by protons initially at rest, at $30°$ . Find the maximum energy of the scattered electron.
1.25scatteringtheory
If $E = 20$ GeV electrons scatter elastically and emerge with $E' = 8$ GeV, find the scattering angle.
1.26scatteringtheory
Find the ratio between the Mott and Rutherford cross-sections for the same particles at the same energy at $90°$ .
1.27sourcestheory
A particle of mass $m$ , charge $q = 1.6\times10^{-19}$ C and momentum $p$ moves in a circular orbit at constant speed in a magnetic field $\mathbf B$ normal to the orbit. Find the relation between $m$ , $p$ and $B$ .
1.28unitstheory
To measure the total $\pi^+p$ cross-section at 20 GeV, a 1 m liquid-hydrogen target is placed between two scintillation counters. Normalised to the same incident flux, $N_0 = 7.5\times10^5$ particles are counted with the target empty and $N_H = 6.9\times10^5$ with it full. Find the cross-section and its statistical error.
- $\rho = 60\ \text{kg m}^{-3}$
- $l = 1$ m
1.29detectorstheory
In the experiment of Chamberlain et al. in which the antiproton was discovered, the antiproton momentum was about $p = 1.2$ GeV. What is the minimum refractive index needed to have the antiprotons above threshold in a Cherenkov counter? How wide is the Cherenkov angle if $n = 1.5$ ?
1.30detectorstheory
Two particles of masses $m_1$ and $m_2$ have the same momentum $p$ . Evaluate the difference $\Delta t$ in the times they take to cross a distance $L$ . With two scintillators measuring $\Delta t$ to 300 ps, how long must $L$ be to separate $\pi$ from $K$ at two standard deviations, if their momentum is 4 GeV?
1.31detectorstheory
A Cherenkov counter containing nitrogen at pressure $\Pi$ sits on a beam of momentum $p = 20$ GeV. The index depends on the pressure as $n - 1 = 3\times10^{-9}\,\Pi\,(\text{Pa})$ . The detector must see the $\pi$ and not the $K$ . In which range must the pressure be?
1.32kinematicstheory
Superman travels down an avenue at high speed. At a crossroads he sees the lights are green and continues, but is stopped by the police, who claim he crossed on red. Assuming both are right, what was Superman's speed?
1.33detectorstheory
For the Cherenkov effect in water ( $n = 1.33$ ), determine (1) the minimum velocity for radiating, (2) the minimum kinetic energy for a proton and for a pion, (3) the Cherenkov angle for a pion of energy $E_\pi = 400$ MeV.
1.34detectorstheory
A threshold Cherenkov counter contains N $_2$ at a variable pressure, with $n = 1 + 3\times10^{-9}\pi$ (pascals). A beam of $\pi^+$ , $K^+$ and protons, all of the same momentum, crosses it. Knowing that the $\pi^+$ are above threshold for $\pi \ge 5.2\times10^3$ Pa, (a) find the momentum, (b) the minimum pressure at which the $K^+$ radiate, (c) the same for the protons.
1.35sourcestheory
(1) What is the maximum energy of a cosmic-ray proton that remains confined in the Solar System ( $R = 10^{13}$ m, $B = 1$ nT)? (2) What is it for the Milky Way ( $R = 10^{21}$ m, $B = 0.05$ nT)?
1.36kinematicstheory
Portable neutron generators use $d + t \to n + {}^4$ He with deuterons accelerated to $T_d = 130$ keV. (1) Calculate the neutron kinetic energy. (2) With an isotropic production rate $I = 3\times10^{10}\ \text{s}^{-1}$ , what is the neutron flux at $R = 1$ m? (3) Tagging the neutron by detecting the $\alpha$ , what time resolution is needed to locate a scattering nucleus to $\Delta z = 5$ cm?
- $m_d = 1875.6$ , $m_t = 2808.9$ , $m_\alpha = 3727.4$ , $m_n = 939.6$ MeV
1.37scatteringtheory
Neutrons of a few MeV are to be detected in a TPC containing $^{40}$ Ar. If the energy is low enough the nucleus scatters coherently, as a single object. Taking a nuclear radius $R_A = 4$ fm, what is the minimum neutron kinetic energy needed to resolve the nuclear structure, and what is the maximum recoil energy of the nucleus at that limit?
- $R_A = 4$ fm
- $m_{\text{Ar}} = 37.2$ GeV
1.38kinematicstheory
Two photons of energies $E_1 > E_2$ collide head on. If $\gamma_2$ comes from a laser of wavelength $\lambda = 694$ nm, what is the minimum $E_1$ to produce an $e^+e^-$ pair? Compute the CM velocity at threshold as $1-\beta$ . What is the mass of the two photons if they move in the same direction?

Where this goes next

Chapter 1 is finished. Chapter 2 stops building tools and starts using them: the discovery of the positron, the muon, the pion and the strange particles, all of it done with the cloud chambers, emulsions and counters of §1.13.

✅ Check yourself — the toolkit, before you use it

0/6 answered · 0 correct

1.Fourteen formulas P1.2–P1.15 look like a lot to memorise. What is the pattern that makes them one formula?
2.Problem 1.6 asks for the threshold to make a particle of mass m in pp collisions. Why is the collider answer E* = m_p + m/2 so much better than the fixed-target E_p = m_p + 2m + m²/2m_p?
3.Problem 1.16 gives exactly 90° between two equal balls after an elastic collision, but Problem 1.17 gives 86° for two protons at 3 GeV. What changed?
4.Problem 1.7 finds that a cosmic-ray proton above ~7×10¹⁹ eV can photoproduce a pion on a 1 meV microwave-background photon, with an attenuation length of a few Mpc. What does that imply?
5.Problem 1.20 finds a π⁰ from stopped-pion charge exchange travels 5.3 nm in a lifetime. What is the practical consequence?
6.Equation (P1.9) is printed with 2m_a m_c where the derivation requires 2E_a E_c. How would you have caught this without an external source?

What you were supposed to take away

The two-body toolkit

The problems

Further reading

Where this goes next

✅ Check yourself — the toolkit, before you use it