Transcript 252b Lecture 6: Passage of Particles Through Matter

Detecting Particles:
How to “see” without seeing…
Interactions of Particles with Matter:
Electromagnetic and Nuclear Interactions
Prof. Robin D. Erbacher
University of California, Davis
References: R. Fernow, Introduction to Experimental Particle Physics, Ch. 2, 3
D. Green, The Physics of Particle Detectors, Ch. 5,6
http://pdg.lbl.gov/2004/reviews/pardetrpp.pdf
Interactions of Particles with Matter
The fact that particles interact with matter allows us to
measure their properties, and reconstruct high energy
interactions.
Dominant interaction is due to the electromagnetic
force, or coulomb interactions.
Ionization and excitation of atomic electrons in matter
are the most common processes.
Radiation can become important, particularly for
electrons.
The nuclear interactions play a less significant role
Elastic Scattering
Incoming particle: charge z
Target nucleus: charge Z
Elastic cross section:
 
d
1
2 2 2 me c
 4z Z re   4
d
 p  sin  2
2
Rutherford Scattering Formula
(valid: spin 0, small angles  low )
Cross section diverges for <>=0.
For very small angles (large impact b),
get screening:
 
d
1
2 2 2 me c
 4z Z re  
2
2
d
 p   2   min

2
At =0 cross section goes to a constant.
Elastic Scattering Cross Section
We can estimate min by localizing the incident trajectory to
within x ~ atomic radius ra for a chance at scattering.
Incident momentum uncertain by p ~ /ra and  ~ / pra .
Using atomic radius:
a0
ra  1 3 
1
Z 2 m cZ2 3
We find the minimum angle:
1 mc


min  Z 2 3
p
Rutherford scattering breaks down when becomes
comparable to rn, nucleus size.

Similar to diffraction:
Integrating
from to , we get:
d

 
sin dd
  
where r  1 r A
1
max
rn
prn
n
3
2 e
Total elastic xs falls off as .
d
2
2
3 

Z1Z 2
2
2
 re 4
 ~ 
  
Multiple Scattering
For any observed angle  of a particle, we don’t know if it
underwent a single scattering event or multiple small angle
scattering. We determine distns for processes, so the
probability of a process resulting in angle can be found.
Expectation of  :
d
 02  Ln a   2
d
d
4  2 m 2c 2 L

Z1 2 2

 p X0
Approximation:
1
0 
p
L
X 0 
X0 = “radiation length”
Detecting Charges Particles
Particles can be detected if they deposit energy into matter.
How do they lose energy in matter?
Discrete collisions with the atomic electrons of absorber.
[Collisions with nuclei not so
important (me<<mN) for energy loss]

dE
d
   Ne E
d
0
dx
dE
If
Ne: electron density
 , k are in the right range, we get ionization.

Classical dE/dx
Charged particles passing through matter have collisions
with nuclei and electrons in atoms. Momentum kick
obtained from coulomb field can be written as (Fernow 2.1):
2z1z2e
p 
v1b
2
If m=me and z2=1 for e,
M=Amp and z2=Z for n:
Incident particle of mass M,
charge z1e, velocity v1. Matter
particle of mass m, charge z2e.
p

E 
1
E(electrons) Z  Z 2 



  4000
E(nucleus) me 
2Zm

p 
2
2z12z22e 4
 2 2
2m
v1 b m
(For Z electrons in
an atom with A~2Z)

Total energy lost by incident particle per unit length:

dE 4n e z12e 4 m v13 2

ln 2
2
dx
m v1
z1e 
 characteristic orbital frequency for
the atomic electron. This classical form
Is an approximation.
Range of Particles
When electromagnetic scattering dominates as a
source of energy loss, a pure beam of charged particles
travel about the same range R in matter.
Example: A beam of 1 GeV/c protons has a range of
about 20 g/cm2 in lead (17.6 cm).
The number of heavy charged particles in a beam
decreases with depth into the material. Most ionization
los occurs near the end of the path, where velocities
are small. Bragg peak: increase in energy loss at end
of path.
0
1
R(E)  
dE
E  dE dx
Mean Range: depth at which
n
R(E)

(E
/
E
)
1/2 the particles remain.
0
Real Ionization Energy Loss
dE
.
dx
Want average differential energy loss
Energy loss at a single encounter with an electron:
pe  Fc t,

ze 2
2b
Fc  2 and t 
b
v
pe 

E 
2
2z 2e 4
 2 2
2m
v b me

Number of
encounters
proportional to
electron density
in medium:
Z
Ne  NA 
A
Introduce classical
electron radius:

2re2 me c 2 z 2 1
b2
2

 2mec 2 2 2 max
dE
 
2
2 2 Z 1 1
2
 4N A re mec z
ln
T    
2 2
2
dx
A
I
2
 
Full-blown Bethe-Block
Bethe-Bloch Formula for Ionization
High energy charged
particles kick off electrons in
atoms while passing thru:
Ionization energy loss.
Characterized by:
• 1/2 fall off (indep of m!)
• predictable minimum
• relativistic rise
The Point: High energy
particles lose energy slowly
due to ionization, so they
leave tracks….
Many kinds of
tracking detectors!
Energy Loss Distributions
• In a thin detector, a
charged particle
deposits a certain
amount of energy
(ionizes a certain
number of atoms)
described by a Landau
distribution
• Notice nice “gap”
between zero charge
and minimum, and
long tail on high side.
Landau Distribution
Plot of signal pulse height made
using 300-m-thick Si strip detector
Silicon Tracking Detectors
• In the past 20 years detectors fabricated directly on
silicon wafers have become dominant for inner tracking
σx = ~10 μm
• large (~200V) reverse bias applied
• ionization e/h drift rapidly, inducing signal (25k e)
Silicon Vertex Detectors
Silicon Pixel Detectors
• we are presently building a detector for the LHC
experiment CMS which uses 100x150 micron pixels
Electrons and Positrons
Electrons and positrons lose energy via ionization just as
other charged particles.
However, their smaller masses mean they lose significant
energy due to radiation as well:
• Bremsstrahlung
• Elastic scattering
• Pair production and electromagnetic showers
Since they have similar electromagnetic interactions, use
electron as primary example.
Ionization: One difference is Bhabha scattering of e- and
e+ instead of e- e- Moller scattering. Bethe-Block using
both cross sections is similar, so to first order, all singlycharged particles with ~1 lose energy at ~ same rate.
Bremsstrahlung
The dominant mechanism for energy loss for high energy
electrons is electromagnetic radiation.
Synchrotron radiation: For circular acceleration.
Bremsstrahlung radiation: For motion through matter.
Time-rate of energy loss
depends quadratically on
acceleration:
dE 2e 2  2
  3 a
dt 3c 
d
e 4 2mec  re2 Mv12 2
 5 Z1 Z 2   ln
dk
c
k
Mv1  k
2
2


Semiclassical calculation:
Bremsstrahlung cross section
for a relativistic particle.
Positron Annihilation
In almost all cases, positrons that pass through matter
annihilate with an electron, to create photons:
e   e    
Single photons are possible if the electron
is bound to a nucleus… this occurs at only
20% the rate for two photons.

A high energy positron will lose energy
by collision and radiation, until it has
a low enough energy to annihilate.
Positronium: e+ and e- can form a
temporary bound state, similar to
the hydrogen atom.
Photons and Matter
Interactions for charged particles: small perturbations.
Energy is small
• slight change in trajectory
Number of incident particles remains basically unchanged.
 Photons: large probability to be removed upon
interaction.
Beam of photons: number N removed from beam while
traversing dx of material is dN = Ndx. The beam intensity
is then
 x
I(x)  I0e
 is the linear attenuation coefficient, dependent on the
material.
Photon Interactions
• low energies (< 100 keV):
Photoelectric effect
• medium energies (~1 MeV):
Compton scattering
• high energies (> 10 MeV):
e+e-
pair production
• Note that each of these
processes leads to the ejection of
electrons from atoms!
• “electromagnetic showers”
PhotoElectric Effect
Interaction between
photon and whole atom.
Photons with energies
above the Work Function,
or binding energy of an
electron, can eject an
atomic electron, with
kinetic energy T:
T  h W
Einstein: Quantized
photon energies
Feynman: QED!
Compton Scattering
Compton Effect:
Scattering an incident photon
with an atomic electron.
Wavelength difference
between incoming and
outgoing photon:
  0  h /mc (1 cos  )
Frequency of Scattered
Photon:

0
1 (1 cos  )
where  =  0 /mc 2
Conversions (ugh!)
Otherwise known as pair
production. Large background: fake
real electron/positron signals.
Total cross section increases
rapidly with photon energy,
approximately proportional to Z2.
Comparing pair production with
bremsstrahlung:
 pair  7 9  brem
Total mass attenuation coefficient:
 NA
NA

 tot 
( pE  Z comp   pair )
A
  A
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Electromagnetic Showers
• a beam of electrons impinging on solid matter will
have a linear absorption coefficient of 1/X0
• this process repeats, giving rise to an e.m. shower:
• the process continues until the resulting photons
and electrons fall below threshold
• so how do we get some sort of signal out?
• ultimately we need ionization
Will discuss more when we talk about calorimetry…
Radiative  Energy Loss
Optical behavior of medium:
characterized by complex dielectric constant
Sometimes instead of ionizing an atom or exciting matter,
the photon can escape the medium: (transition, C, etc).
Nuclear Interactions
Next Monday, finish up with nuclear interactions.
Strong Interactions
Weak Interactions