The Sources of High-Energy Particles · Elementary Particle Physics

One formula sets the size of every circular machine ever built: momentum is 0.3 × field × radius. Fields stop at a few tesla, so higher energy has only ever been bought with more circumference.

🎯 Why this matters

That is why the field’s future is argued in kilometres. Superconducting magnets raised the ceiling once, and every gain since has come from digging a longer tunnel rather than from a better idea.

You need particles at GeV energies. There are exactly two ways to get them: wait for the sky to send some, or build a machine. The field did the first for forty years and has done the second ever since, and this section is both.

Cosmic rays: the free beam

Nobody was looking for them. In 1910–11 D. Pacini measured how fast a charged electrometer discharged — on land, at sea 300 m from shore, and 3 m underwater — and found a source of ionisation that was not the ground’s radioactivity. In 1912 V. F. Hess took electrometers up in balloons to 5.2 km and found the flux rising with altitude, doubling by the top of his flight. The radiation was coming from above.

all-particle cosmic-ray spectrum (approximate)

Twelve decades of energy against roughly thirty of flux — the widest dynamic range in any plot in this book. The two kinks (the 'knee' and the 'ankle') are where the acceleration mechanism or the source population changes. The LHC's entire 14 TeV, converted to what a fixed-target experiment would need, lands at 0.1 EeV — comfortably inside the cosmic-ray spectrum, but at a flux of about one particle per square kilometre per century, which is why the Pierre Auger Observatory covers 3000 km².

At a few GeV, where the flux is largest and where the discoveries of Chapter 2 were made, the composition arriving at the top of the atmosphere is 85 % protons, 12 % α particles, 1 % heavier nuclei and 2 % electrons.

What happens on the way down

A primary proton hits an air nucleus after about one collision length ( $\lambda_0 = 750$ m at n.t.p., from §1.11) and starts a cascade that splits into three components with completely different fates.

Fig. 1.14One primary, three fates — and every one of them is an application of §1.11. The soft component is bremsstrahlung and pair production alternating; the hard component survives because muons are 207 times too heavy to radiate; the neutrinos escape because the weak interaction is all they have.

💡 What this really says — why muons reach the ground at all

A muon created at 15 km with a 2.2 μs lifetime should travel about 660 m before decaying — it should never arrive. It does, in quantity, and the reason is §1.1: at γ ≈ 20 its lifetime in our frame is 44 μs and it covers 13 km. Cosmic-ray muons at sea level are a standing experimental proof of time dilation, running continuously, free, everywhere.

Accelerators: making the beam instead

Cosmic rays are free but you cannot choose their energy, their species or their arrival time. From 1952 the field switched to machines, and one design has dominated ever since: the synchrotron.

🛠️ A synchrotron, schematically

Click a numbered marker for what that piece does.

Four components, and every accelerator in this book is a variation on them. The physics is entirely in callouts 1 and 3; callout 2 is what makes it work in practice.

\htmlClass{t-p}{p}\,[\text{GeV}] = 0.3\,\htmlClass{t-B}{B}\,[\text{T}]\;\htmlClass{t-R}{R}\,[\text{m}]

(1.110)

The single most-used engineering formula in the subject — it sizes every ring and every spectrometer magnet in the book.

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.

🔢 Worked example — the formula reproduces real machines

Rearranged as

R = p/(0.3B)

Machine	p (GeV)	B (T)	R from the formula	Reality
CERN PS	28	1.2	78 m	70 m
SPS	450	1.8	834 m	~740 m
Tevatron	980	4.4	743 m	~750 m
LHC	7000	8.33	2803 m	2804 m

The agreement for the LHC is exact to four figures, because that is precisely how the machine was specified: fix the tunnel (inherited from LEP), fix the field (what niobium–titanium can do at 1.9 K), and the beam energy follows.

Colliders: stop wasting the energy

§1.4–1.5 showed why. Only $\sqrt{s}$ is available for making particles, and a fixed target throws most of the beam energy into the motion of the centre of mass. The equivalence is

E^* = \sqrt{\frac{m_t E_b}{2}} ,

so the SPS’s 450 GeV beam on a fixed target is worth only 14.5 GeV per beam in a collider — and the same magnets, used to collide two beams instead, gave $\sqrt{s} = 540$ GeV.

beam energy (log₁₀ of GeV)3

beam = 1000 GeV
fixed target (on a proton): √s = 43.339 GeV
collider (two such beams): √s = 2000 GeV
collider is 46.15× better at the same beam energy

collider: √s = 2E
fixed target: √s ≈ √(2m_p E)

On a log–log plot the collider line has slope 1 and the fixed-target line slope ½. To gain a factor of ten in √s you buy ten times the beam energy in a collider — or a hundred times at a fixed target. That gap is the entire reason colliders exist.

Put differently: to match the LHC's 14 TeV with a fixed-target machine you would need a beam of 104.4×10⁶ GeV = 0.104 EeV — an energy only the rarest cosmic rays reach.

🛠️ A collider

Click a numbered marker for what that piece does.

The ISR (1971) was the first proton collider; ADA (1961, 250 + 250 MeV) was the first particle–antiparticle one. The LHC is the same diagram with two rings, 8.3 T magnets and 27 km of circumference.

\htmlClass{t-L}{\mathcal{L}} = \htmlClass{t-f}{f}\,\frac{\htmlClass{t-n}{n_1 n_2}}{\htmlClass{t-S}{\Sigma}}

(1.112)

Everything a machine builder can control, in one line.

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 machines this book will refer to

Machine↕	Year↕	Type↕	Energy↕	What it is remembered for
Cosmotron (BNL)	1952	proton synchrotron	3 GeV	The first; the end of the cosmic-ray era.
Bevatron (Berkeley)	1954	proton synchrotron	7 GeV	Built specifically to exceed antiproton threshold — and it did (§2.7).
PS (CERN) / AGS (BNL)	1960	proton synchrotron	30 GeV	The ν_μ discovery, the Ω⁻, CP violation.
ADA (Frascati)	1961	e⁺e⁻ collider	0.25 + 0.25 GeV	Touschek's proof that a particle–antiparticle collider works at all.
ISR (CERN)	1971	pp collider	31 + 31 GeV	The first hadron collider.
SPS (CERN)	1976	proton synchrotron	450 GeV	Converted to a pp̄ collider in 1981 — see below.
SppS (CERN)	1981	pp̄ collider	270 + 270 GeV	W and Z discovered, 1983 (§9.7). Rubbia and van der Meer, Nobel 1984.
Tevatron (FNAL)	1987	pp̄ collider	1 + 1 TeV	The top quark, 1995 (§4.10).
LEP (CERN)	1989	e⁺e⁻ collider	up to 105 + 105 GeV	Precision electroweak; three neutrino families (§9.9).
HERA (DESY)	1991	ep collider	30 e × 920 p GeV	The highest-resolution microscope ever built — proton structure (§6.2).
LHC (CERN)	2010	pp collider	up to 6.8 + 6.8 TeV	The Higgs boson, 2012 (§9.15).

Notice the pattern: a fixed-target synchrotron is often later converted into a collider using the same tunnel and magnets (SPS → SppS, LEP tunnel → LHC). The civil engineering is the expensive part and it outlives several machines.

⚙️ Engineer’s bridge — an accelerator is three control loops

Almost nothing that makes these machines work is particle physics. Three of the ideas above are feedback, and each has a name you already use.

Phase stability is a phase-locked loop. The RF cavity gives a push that depends on when the particle arrives. Choose the synchronous phase on the rising edge and a particle arriving early gets less energy, so it takes a longer orbit and arrives later next turn; one arriving late gets more and catches up. That is a restoring force in the phase variable, and the beam executes damped oscillations about the synchronous phase — synchrotron oscillations, which are the loop’s ringing. The bunch structure of every beam in this book is not a design choice: it is the capture range of a PLL, and particles outside it are simply lost.

Alternating-gradient focusing is parametric stabilisation. A quadrupole focuses in one plane and defocuses in the other, so no single magnet can confine a beam — the equilibrium is a saddle. Alternate them fast enough and the motion is stable anyway, exactly as an inverted pendulum on a rapidly oscillating pivot stands up (the book’s own image is a stick balanced on a moving hand). The formal statement is one you have met: write the one-period transfer matrix $M$ of the lattice, and the motion is stable if and only if

|\operatorname{Tr} M| < 2,

which is the condition that $M$ ‘s eigenvalues sit on the unit circle rather than one of them growing each turn. It is the same eigenvalue test as the stability of any discrete-time loop.

Stochastic cooling is adaptive noise cancellation. Van der Meer’s technique puts a pickup electrode on the beam, an amplifier, and a kicker further round the ring. The pickup measures how far a sample of particles deviates from the ideal orbit; the kicker corrects it. Two things make it work, and both are engineering:

The correction must arrive before the particles do. The signal takes the chord across the ring at $c$ through a cable while the beam takes the arc at slightly less than $c$ . The geometry buys you the latency budget.
Bandwidth is the whole resource. You cannot resolve one antiproton, so each measurement averages $N_s$ of them and each kick removes only a $1/N_s$ slice of the spread. Wider bandwidth means shorter samples, fewer particles per sample, and faster cooling: the ideal floor is $\tau \simeq N/2W$ . For $N = 10^{11}$ antiprotons and $W = 2$ GHz that is 25 seconds — and real machines are far slower, because amplifier noise and imperfect mixing both fight you, exactly as they do in any adaptive filter.

Rubbia and van der Meer shared the 1984 Nobel Prize. Rubbia’s half was the physics; van der Meer’s was a control loop.

Where it breaks: the three loops are not independent and cannot be tuned in isolation, because they share one plant — the beam — and each one’s actuator is another’s disturbance. Stochastic cooling has a hard limit no control textbook would predict from the block diagram: it can only correct a sample it can resolve, so the achievable cooling rate is set by the system bandwidth divided by the number of particles, and adding intensity makes the loop worse. And the plant cannot be taken offline to be characterised: 362 MJ of stored beam means a badly tuned loop does not oscillate, it destroys the magnet.

Reproduce it

import numpy as np
c, mp = 299792458.0, 0.938272
print(f"p[GeV] = {c/1e9:.4f} B[T] R[m]   (the book's Eq. 1.110 rounds this to 0.3)")

print("bending radius from the formula, against the real machines:")
for name, p, B in (("PS", 28, 1.2), ("SPS", 450, 1.8), ("Tevatron", 980, 4.4), ("LHC", 7000, 8.33)):
    R = p / (0.2998 * B)
    extra = "   (actual: 2804 m)" if name == "LHC" else ""
    print(f"  {name:9s} p={p:5g} GeV  B={B:.2f} T -> R = {R:7.1f} m{extra}")

print("fixed-target beam -> equivalent collider energy per beam, E* = sqrt(m_p E_b / 2):")
for name, Eb in (("Bevatron", 7), ("SPS", 450), ("LHC beam", 7000)):
    print(f"  {name:9s} {Eb:5g} GeV -> E* = {np.sqrt(mp*Eb/2):6.2f} GeV")

g, tau = 20.0, 2.1969811e-6                       # cosmic-ray muon
print(f"a cosmic-ray muon at gamma = {g:.0f}: lab lifetime {g*tau*1e6:.1f} us, "
      f"range {g*tau*c*np.sqrt(1-1/g**2)/1e3:.1f} km")
print(f"  (without time dilation it would travel only {tau*c/1e3:.2f} km "
      f"and never reach the ground)")

N, W = 1e11, 2e9                                  # antiprotons, cooling bandwidth
print(f"stochastic cooling floor tau = N/(2W): N={N:.0e}, W={W/1e9:.1f} GHz "
      f"-> {N/(2*W):.1f} s")

prints

p[GeV] = 0.2998 B[T] R[m]   (the book's Eq. 1.110 rounds this to 0.3)
bending radius from the formula, against the real machines:
PS        p=   28 GeV  B=1.20 T -> R =    77.8 m
SPS       p=  450 GeV  B=1.80 T -> R =   833.9 m
Tevatron  p=  980 GeV  B=4.40 T -> R =   742.9 m
LHC       p= 7000 GeV  B=8.33 T -> R =  2803.0 m   (actual: 2804 m)
fixed-target beam -> equivalent collider energy per beam, E* = sqrt(m_p E_b / 2):
Bevatron      7 GeV -> E* =   1.81 GeV
SPS         450 GeV -> E* =  14.53 GeV
LHC beam   7000 GeV -> E* =  57.31 GeV
a cosmic-ray muon at gamma = 20: lab lifetime 43.9 us, range 13.2 km
(without time dilation it would travel only 0.66 km and never reach the ground)
stochastic cooling floor tau = N/(2W): N=1e+11, W=2.0 GHz -> 25.0 s

🔑 If you remember only three things

The sky was the first beam, and it is uncontrolled. You cannot choose the energy, the particle or the moment, which is exactly what a machine buys and why the field stopped waiting.
Time dilation is a design parameter here, not a curiosity. A muon beam exists at all because γτ can be made long enough to build apparatus around.
The hard part is keeping the beam, not making it. Getting one particle fast is easy; holding 10¹¹ of them in a bunch for hours is what the machinery is actually for.

Where this goes next

§1.13a–d is what you put at the end of the beamline.
Chapter 2 is the cosmic-ray era in full, before any of these machines existed.
§9.6–9.7 is the SppS story: how converting one machine and inventing stochastic cooling produced the W and Z.
§9.14 is the LHC and its detectors, where the 40 MHz of this section’s luminosity formula becomes the trigger problem.
The discovery timeline puts these machines on one axis with everything they found, and shows the three eras the instruments define.

✅ Check yourself — where the particles come from

0/5 answered · 0 correct

1.A muon is created 15 km up with a proper lifetime of 2.2 μs. Travelling at nearly $c$ it should manage about 660 m. Why do muons dominate the cosmic radiation at sea level?
2. $p[\text{GeV}] = 0.3\,B[\text{T}]\,R[\text{m}]$ . The LHC holds 7 TeV with 8.33 T magnets. What bending radius does that require, and what does the agreement tell you?
3.A magnetic quadrupole focuses in one plane and defocuses in the other. How does a synchrotron get net focusing?
4.The SPS ran as a 450 GeV fixed-target machine and was then converted to collide 270 GeV on 270 GeV. Why was that worth doing?
5.The all-particle cosmic-ray spectrum reaches 10²⁰ eV. Why do we build accelerators at all, when nature offers higher energies for free?