Accelerate This!
Using the forces of
electricity and
magnetism
to make tiny things
go really fast
Daniel Friedman, St. John’s School
Accelerating a charged particle
Remember ‘opposites attract’ and
‘likes repel’?
Let’s call it
Coulomb’s
Law!
Qq
F k 2
r
Accelerating a charged particle
The force on charged particle q is due to the
charge Q of another nearby particle;
For two charges
of the same
sign:
Qq
F k 2
r
Another way to look at this uses
the idea of an electric field
If we only know one of the
charges (Q), we can still
calculate the force per unit of
charge some unknown q
would ‘feel’:
F
Q
E k 2
q
r
Still another way:
Voltage = work/unit charge
To move a charge in the presence of a repulsive
Coulomb force, we have to do some work. But
we can express this in terms of the E field:
F d
Work
Ed

q
unit charge
Voltage is sometimes called
‘electromotive force’ but it is better
described in terms of energy
Work
Ed
 voltage
unit charge
We also use the term ‘potential difference’
as a synonym for voltage.
A charge crossing through a
potential difference gains kinetic
energy
The energy measure of particle
physics is the electron volt (eV),
defined as the energy of a single
electron across a potential
difference of 1 Volt:
DKE = qV
An eV is tiny!
1 eV = 1.6x10-19 Joule
keV = 103 MeV = 106
GeV = 109 TeV = 1012
Kilo – Mega – Giga – Terra
A billion eV was also known as BeV,
which led to the term ‘Bevatron’.
Want high energy?
You’ll need a lot of batteries
If you put 600 billion D cell
batteries in series, end to
end, you’d get 900 billion
volts – and each electron
would have 900 GeV!
But your circuit would be
23 million miles long!
Some typical energies
keV=103
MeV=106 GeV=109 TeV=1012
Visible light photon: 1.5-3.5 eV
Ionize atomic hydrogen (free the proton!): 13.6 eV
Your TV set: 20 keV
Medical X-Rays: 200 keV
Natural radioactivity (a+2, b-, g): 2-5 MeV
Some unusual
energies
FermiLab Tevatron: 900 GeV
CERN’s LHC (under construction): 7 TeV
A Roger Clemens fastball: 7 x108 TeV
(but that’s spread over a lot of particles!)
Highest energy cosmic ray showers: 109 GeV (106 TeV)
Natural Radioactivity
Ernest Rutherford
used naturally
occurring alpha
particles with
energies of
approximately
5 MeV to discover
the nucleus.
At this energy level, all
Rutherford could detect were
the collisions between ‘solid’
objects
But to see inside the nucleus,
we need higher energy!
Welcome to
Big Science:
FermiLab
main
accelerator ring
over 4 miles
around;
900 GeV
Particle accelerators!
SLAC is 3 km
long. Initially
18-20 GeV;
upgraded to
50 GeV
Early
Accelerators
Using a very high DC
voltage (large E field) to
accelerate an electron
across a small gap.
Cockroft-Walton
Accelerators
John Cockroft and Ernest
Walton, students of
Rutherford at Cambridge,
were leaders in developing
this technology.
CW accelerators produce very
high DC voltages
across a small ‘gap’
+ ion source
high
voltage
DC
or rectified
AC
energy gain = qV
(max ~200 keV)
The high voltage is obtained
by means of a ‘diode-capacitor
ladder’
Popular science fiction
liked the look of
these early accelerators
Robert Van deGraaf
uses a moving belt
to build charge by friction
1931: A giant Van deGraaf in an
old dirigible hanger
Each sphere was at a
potential of 750 kV for a
total potential difference of
1.5 MV!
The observation
labs were inside
the spheres!
Question:
Why can the observer sit inside
the sphere without harm?
A big enough Van de Graaf can
produce 10 MeV: the radius of
the sphere is the voltage limit
Linear Accelerators
+
e-
-
Copper rings are used as
the accelerating field
plates.
Linear Accelerators
-
+
e-
The field polarity is
switched by a radio
frequency oscillator.
Acceleration occurs due
to the E field across each
gap:
F = Eq = ma
Linear Accelerators
+
e-
The field polarity is
switched by a radio
frequency oscillator.
Acceleration occurs due
to the E field across each
gap:
F = Eq = ma
Linear Accelerators
e-
The disks act as
‘collimators,’
blocking any
electrons that are
not going in the
direction of the
beam.
Interactive linac
The biggest linacs
use of many
thousands of gaps
and generate 100’s of
MeV – requiring a
very straight hole in
the ground!
Some medical devices are small
linear accelerators
You might see one
the next time you go
to the dentist!
Linacs deliver ‘bunches’ of
electrons
Very high-grade
vacuum:
Internal pressure
10-12 torr!
(1 atm=760 torr)
Let’s introduce
the magnetic force
A charged particle with velocity v in the presence
of an external magnetic field of strength B
experiences a force
if the mag field has
a component that is
perpendicular to the
motion of the charge.
Sine=opposite/hypoteneuse
The magnetic force is
perpendicular to the velocity of
the charge
A perpendicular force can only change
the direction of motion, not the speed.
If the external B field
is constant,
the particle
moves in a circle.
The magnetic force is given by
F  qv  B
| F |  qvB sin 
Known as the ‘cross product’.
The direction of the magnetic
force is given by the
‘right hand rule’
Fingers along the B field,
thumb in the direction of
motion of a positive
charge,
palm points in the
direction of the force.
Putting these forces together,
F  q ( E + v  B)
the ‘Lorentz force.’
It is vital to note that E and B are always perpendicular.
Cyclotrons: External B field
creates a spiral path
Relatively low
potential difference
across the gap
between opposing
‘D’ ’s.
Acceleration each time
gap is crossed.
Top view, showing one of the ‘D’
electrode cavities
Cyclotrons: particles accelerate
around a spiral path
As particles gain
energy, they spiral out.
mv 2
F  qvB sin  
r
mv
r
qB
Cyclotron Resonance
Radius increases with
increasing speed; but the
time required to complete
½ circle remains constant.
The frequency of the E
field may therefore be
kept constant.
Cyclotrons
First cyclotron
4 inches in diameter
80 keV
1MeV Lawrence
Cyclotron (1932)
11 inches in diameter
Cyclotrons
President
Eisenhower
inspects
Columbia’s 400
MeV Cyclotron
(1950)
Cyclotrons
Modern 250 MeV
Cyclotron might be part
of a PET scan device or
cancer treatment facility
Cyclotrons: problem 1
Higher energy
requires more turns
around the ring.
This means a bigger
ring – and a bigger,
more expensive
magnet.
Cyclotrons: problem 2
Iron core magnets can
withstand a maximum
of about 2 Tesla before
the iron starts to turn
purple (and burn up).
The maximum B field
strength thus limits the
energy at the outside
edge.
mv  rqB
1 2 1
mv  rqvB
2
2
The Synchrotron:
particle racetracks!
Smaller magnets are
used only to steer and
focus the particles.
Changing the B field
strength to compensate
for increasing velocity
maintains a fixed
radius.
Synchrotrons
Pushing the particle
at just the right
time increases
speed – just like
pushing a child on
a swing.
Accelerating protons
Accelerating anti protons
Oppositely
charged
particles turn in
opposite
directions in
the same
beamline.
What’s the direction of the B field here?
A fact that is clearly marked on
the road signs
Synchrotrons
The Cosmotron at BNL: First GeV accelerator!
Go Big Red!
6 GeV under the
football field at Cornell.
The 192 magnets are
each only 3 m long.
Synchrotrons
6 GeV under a town in
Germany
Synchrotron
Radiation
Electrons accelerating in the external magnetic
field give off visible ‘synchrotron radiation.’ This
lost energy must be supplied by the E field.
Protons do not radiate as much due to their higher
mass.
Magnets steer the beam
Big B fields require large
currents, generating lots of heat.
There are 42500 miles of wire
(8x10-6 m diameter) in each
magnet in use at Fermi.
Quadrapole Focusing Magnets
A particle near the
center of the beam
experiences 0 force.
Particles drifting
vertically out of line
are forced back to the
center.
Quadrapole Magnets for
Focusing
Paired quad magnets:
the first focuses the
beam vertically, the
second horizontally.
Particle
beams can be
extracted for
fixed target
experiments
A focusing magnet can ‘nudge’
the two beams together in a
collider experiment
Experiments compared
The full system at Fermi starts
with linear accelerators
FermiLab’s CW accelerator
obtains 750 keV by using a
rather large diode-capacitor
ladder.
For comparison, your TV is
a 20 keV accelerator.
The 400 MeV
nd
2
stage linac
E = 3 Mega volts/meter;
130 m long.
If the linac had to produce
the entire 900 GeV, it
would have to be 182
miles long!
Inside the linac
The ‘booster’ synchrotron
(940 ft diameter) produces 8 GeV
Booster
Particles in the booster
go around 16000 times,
gaining 500 keV per
turn.
Main ring
Injector
The complete accelerator system
at Fermi uses a combination of
synchrotrons
Tevatron ring produces
900 GeV using
superconducting magnets
Injector Ring takes in 8
GeV from the booster and
outputs 150 GeV
Injector tunnel
Injector magnets
(below)
Recycler magnets
(above).
Inside the tunnel
Older Main Ring
(above) ran on
conventional magnets.
Tevatron ring (below)
produces 900 GeV
using superconducting
magnets
Beamline must be a very high
quality vacuum (10-6 atm)
Stray particles can
degrade the beam.
Very tricky to make
flexible joints hold
vacuum!
Super conducting magnets
Who is Jessica?
Built at Fermilab’s
magnet shop, these
bad boys produce a
field of over
4 Tesla –
theoretically
capable of 15 T!
Super conducting magnets
At full power, the
magnets represent
an inductance of
36 henries, storing
288 megaJoules of
energy in the B
field.
It can take up to 2
minutes for this
field to shut off.
Super conducting magnets
Since they are
superconducting,
the cryo system
only requires 13
megawatts of
electricity to keep
them cool.
Conventional
magnets would need
over 600 MW just
for cooling!
Need a good reason to study
computer science?
Colliding antiprotons with protons, each
at 900 GeV:
Center of mass energy = 1.8 TeV!
Trick question:
Center of mass momentum = ?
Big machines = High energy
Protons run around the
accelerator in a ‘bunch,’
consisting of about 2x1013
particles, with total mass ~
4x10-14 kg.
At 900 GeV per proton, this
is the same KE as a 1000 kg
object moving at 65 m/s!
Who,
me?
The collider that wasn’t: SSC
20 TeV!! 87 km around!!
The collider that wasn’t: SSC
Magnets produce 6.5 Tesla!!
Coming in 2007: the LHC
In Switzerland (CERN)
24 km around: 14 TeV!
Collision animation
Collisions produce short-lived
new particles
View live events at Fermi
Such as those that existed in the
first milliseconds of the universe!
Animated collisions
Sources
•
•
•
•
•
•
•
•
Lederman: The God Particle
http://www.fnal.gov
http://www.aip.org/history/lawrence/
http://www.lbl.gov/image-gallery/imagelibrary.html
http://www.mos.org/sln/toe/history.html
http://public.web.cern.ch/public/
http://www2.slac.stanford.edu/vvc/Default.htm
http://www.hep.net/ssc/new/repository.html
When tiny things are going this fast,
we need relativistic momentum
Rest energy E0  m0 c 2 and m  g m0
E  mc 
2
m0 c
2
2
v
1- 2
c
2
v
E 2 (1 - 2 )  E0 2
c
2
2
v
v
E 2  E0 2 + E 2 2  E0 2 + (mc 2 )2 2
c
c
E  E0 + m v c  E0 + p c
2
2
2 2 2
2
2 2
Relativistic energy-momentum
formula
E  E0 + p c
2
2
2 2
For objects moving really fast, p2c2 >> E02,
So E = pc
Relativistic momentum
example
E  E0 + p c
2
2
2 2
An electron with v= .95 c:
m0= 0.51 MeV/c2, g =3.2,
p= g m0v = 1.55 MeV/c
If we did not use relativistic momentum,
p=m0v= (0.51 MeV/c2) (.95c)= 0.485MeV/c
An error of 68%!
Remember the radius of the
cyclotron spiral?
mv
r
qB
Since we now can deal with
relativistic momentum,
r
g m0 v
qB
Relativistic energies provide
another nice simplification
E = mc2 = E0+K= m0c2+K
mc2 = gm0c2 = m0c2+K
g = 1+K/m0c2 = 1+K/E0
g
1
2
v
1- 2
c
or v  1 -
1
g
2
c
Relativistic energies: examples
How fast will 100 MeV move an e- ?
g = 1+K/m0c2 = 1+K/E0
For e-, E0 = .51 MeV.
So g = 1+100MeV/.51MeV = 197
v=.99999c
Electron-positron colliders were very
popular!
Relativistic energies: examples
How fast will 100 MeV move a p+?
g = 1+K/m0c2 = 1+K/E0
For p+, E0= 938 MeV.
So g=1+100MeV/.51MeV= 1.11
v= .428c
Its lots harder to accelerate protons,
but proton-antiproton colliders are
now all the rage!
Trick question
How fast will 100 MeV move an n0?
Its lots,lots,lots harder to accelerate
neutrons – they’re not charged!