Transcript Slide 1

Electromagnetic interactions
• Energy loss due to collisions
– An important fact: electron mass = 511 keV /c2, proton mass = 940
MeV/c2, so it is much easier to give an electron a "kick" than a nucleus,
i.e. will be dominated by interactions with the electrons.
• Other types of e.m. interaction,
– bremsstrahlung and creation of electron-positron pairs by high-energy
photons are sensitive to the electric field strength, so the interaction
with the nucleus dominates.
• Cerenkov/Transition radiation
– A third category of interactions is sensitive to bulk properties of the
matter, like dielectric constant. These interactions give rise to
Cherenkov and transition radiation
Physics 70010 Modern Lab
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

Taking into account quantum-mechanical effects and using first-order
perturbation theory the Bethe-Bloch equation is obtained:
2
2
2


d E 2 22
Z

m
c


T
2
2
e
m ax
1 2

4

N
r
m
c
Z
ln 2




A
e
e
1
2
2
dx
A

I


Tmax is the maximum energy transfer to a single electron:

1
2

m
m
2 22
, 
T

2
m
c

1

2
 e e
m ax
e
2
 MM

Hans Albrecht Bethe
Tmax is often approximated by 2me22. re is the classical
electron radius (re = e2 / mec2 = 2.82 x10-13 cm)
(radius of a classical distribution of the electron charge
with electrostatic self-energy equal to the electron mass).
I is the mean ionization energy.
2 22

2
m
c
NB: for high momentum particles T
max
e 
Substituting this and also e2 / mec2 for re gives eq. (2.19) of Fernow

Physics 70010 Modern Lab
Felix Bloch
2
2
2
2


dE2 22
Z

2
m
c


T
(

)C
(

)
2
2 1
e
m ax

4

N
r
m
c
Z
ln 2


 

A
e
e
1
2
2
dx
A

2Z
 I

2
 is the "density correction“:
It arises from the screening of remote electrons by close electrons, which results
in a reduction of energy loss for higher energies (transverse electric field grows
with !). The effect is largest in dense matter, i.e. in solids and liquids.

C is the "shell correction" :
Only important for low energies where the particle velocity has the same order of
magnitude as the "velocity" of the atomic electrons.
For improved accuracy more correction factors need to be added, but the
particle data group claims that the accuracy in the form shown above for
energy loss of pions in copper for energies between 6 MeV and 6 GeV
about 1 %, with C set to 0.
Note that the Bethe-Bloch equation provides only the mean of the
"stopping power", but no information on fluctuations in it
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dE/dx for pions as computed with Bethe-Bloch equation
dE/dx divided
by density 
(approximately
material
independent)
slope due to 1/v2
relativistic rise
due to ln 
 about proportional to ne,
as ne = na Z = NA  Z / A, -> ne ≈ NA  / 2
Physics 70010 Modern Lab
high :
dE/dx
independent
of 
due to
density
effect,
"Fermi
plateau"
From PDG, Summer 2002
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Some phenomena not taken into account in the formula are :
• Bremsstrahlung: photons produced predominantly in the electric field of the
nucleus. This is an important effect for light projectiles, i.e. in particular for
electrons and positrons
• Generation of Cherenkov or transition radiation. Cherenkov radiation occurs when
charged particles move through a medium with a velocity larger than the velocity
of light in that medium. Transition radiation is generated when a highly relativistic
particle passes the boundary of two media with different dielectric constants. The
energy loss is small compared to the energy loss due to exciation and ionization
• For electrons and positrons the Moller resp. Babha cross sections should be
used in the calculation of dE/dx, this leads to small corrections. Fernow
quotes, for  -> 1, Tmax set to 2me22 and without density and shell corrections:
2


dE 2 2Z

2
m
c
2
e

2

N
r
m
c 
2
ln 
3
ln


1
.
95
Electrons:

A
e
e
dx
A
I


2


dE 2 2Z

2
m
c
2
e
2

N
r
m
c 
2
ln 
4
ln


2
Heavy particles:  

A
e
e
dx
A
I




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

Range of stopping particles
For thick enough material particles will be stopped, the range can
be calculated from (M = mass projectile, Z1 = charge projectile):
0
1
R(E)

dE


dE
dx
E
The Bethe-Bloch equation with Tmax approximated by 2me22 can be
written as:
Mc2
dE 2
2
 Z
(v
)Z
EM
 f(v) can be replaced by g(E/M), as : E
1f
1g
dx
12
dE
M
1
R
(
E
)
M

h
E
M


-> The dependency of R  Z12/M on E is
Z
2
2
g
E
M

 MZ
1
approximately material and projectile
1
 (dE/dx)/ is ~ material
independent(
independent)
Two different projectiles
with same energy:
Z2
Z2
aR  bR
M a M b
a
b
Physics 70010 Modern Lab
Z2 M
Ra  b a R
Z2 Mb b
a
6
Most of the energy
deposited at end of track
Fraction of particles
surviving
100 %
Sir William
Henry Bragg
Sir William
Lawrence Bragg
dE/dx
Bragg
curve
Averange
range R
Depth x in material
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Fluctuations in energy loss
– The energy transfer for each collision is
determined by a probability distribution.
– The collision process itself is also a process
determined by a probability distribution.
– The number of collisions per unit length of
material is determined by a Gaussian
distribution
– the energy loss distribution usually is referred
to as a "Landau" distribution. This is a
distribution with a long tail for high values of the
energy loss. The tail is caused by collisions with
a high energy transfer.
Physics 70010 Modern Lab
Lev Davidovich Landau
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From PDG, Summer 2002
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