Download 2 The interaction of energetic particles with material

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The interaction of energetic particles with material
In order to detect particles, they need to interact with (detection) material. Four important
interactions are:
-
the Electro-Magnetic (EM) interaction of a energetic charged particle with matter;
Bremstrahlung
the interaction of photons (gamma’s, X-rays) with matter
Cerenkov radiation
The Electro-magnetic (EM) interaction
In short: if a fast charged particle transverses a thin layer of material, then a force will be
applied between this particle an all charged constituents in the material (fig.1). The energy
transfer may result in charge separation in the form of creation of ion-electron pairs, or
electron-hole pairs. These two processes lead to charge or light signals which can be
amplified and detected.
Charged particle
Energy E
Energy E’
dX
Fig.1: After crossing a layer of material, a charged energetic particle looses energy dE.
Calculation of force between charged particle (i.e. electron) in material and the fast
particle (1) and the particle (2) in the material.
Energy loss due to e.m. interactions, also referred to
as "collisions",
In the laboratory system S' particle 1 is moving, particle 2 is in rest.
Consider a system S in which particle 1 is in rest, so particle 2 is
moving in a static electric field.
E⊥ = cos θ Z1 e / r2 = Z1 e b / r3
Movement particle 2 in S
θ
r
Z1 e
1
x = -v1 t
(energy particle is
assumed to be high,
-> b is assumed to
be constant)
2
b
t = 0: distance between
1 and 2 is minimal
In the laboratory system S' particle 2 is in rest, so there is no effect
from the (time dependent) magnetic field caused by particle 1.
time in S' !
In S': E ⊥' = γE ⊥ = γZ1eb /r 3
electric field
strength
r 2 = b 2 + x 2 = b 2 + v12t 2 = b 2 + γ 2 v12t'2
3/2
E ⊥ = γZ1eb /(b 2 + γ 2 v12t' 2 )
Then: integration along the X-axis. Note that, due to their mass differences, the energy
transfer to (light) electrons is much larger than to a nucleus.
∞
∆p ⊥ =
∫ F dt = ∫
Z 2e γ Z1edt
⊥
−∞
(b
3
2
+ γ 2 v12t 2 )2
∞




t
2Z1Z 2e 2
∆ p = ∆ p ⊥ = Z 2e γ Z1e
=
1
v1b
 b 2 (b 2 + γ 2 v12t 2 )2 
−∞
2
∆E =
energy loss
(∆p)
2m
=
2Z12 Z 22e 4
b 2 v12m
(Particle 2 is in rest -> non-relativistic
calculation of energy loss possible)
Interactions with electron:
m = me, Z2 = 1
∆ E(e)
2Z 2e 4
= 2 12
∆ E(nucleus) b v1 me
Interactions with nucleus with
mass number A and atomic number Z:
m = Amp ≈ 2Zmp, Z2 = Z
2mp
2Z12 Z 2e 4
=
≈ 4000 /Z
b 2 v12 2Z mp Z me
-> Energy loss due to collisions is dominated by interactions with the electrons
(NB: we are comparing interactions with 1 electron to interactions with the nucleus of an atom,
a non-ionized atom has Z electrons)
Then: integration over b for many participating charge carriers (electrons):
We have looked at the interaction between a charged particle and a single
isolated electron or nucleus. In reality we are dealing with the combined effect
of interactions with many electrons (and nuclei). The effect of this was estimated
by N. Bohr. He considered interactions with atoms, each with Z2e electrons.
Particle 1 is passed through the centre of a thin shell of atoms and
the net energy loss is calculated. Then one integrates over b
(the "impact parameter") to obtain dE/dx
e
Number of electrons in shell: ne 2πbdb dx
with ne = number of atoms per cm3
b
∆E =
r
db
Z1 e
dx
4πZ12e 4 n edbdx
bv12m
 2Z 2e 4  b max db 4π n e Z12e 4
b
dE
= 2π n e  1 2  ∫
ln max
=
2
m
v
b min
m
v
dx
b
 e 1 b
e 1
min
We first need to define the minimum and maximum values of b:
3
To find bmax we note that (b 2 + γ 2 v12t 2 )2 determines the size of the
transversal field strength, from this it follows that only during a time of the
order of b/(v1γ) the "collision" takes place. This time should be short relative
to the time characterizing the orbital frequencies of the electrons in the atoms,
otherwise the effect of the binding of the electrons cannot be neglected
(Bohr considered harmonically bound charges for distant collisions, but
numerically the results are about the same as for the approach discussed here).
Define ω to be the characterizing orbital frequency, then: bmax = γv1/ ω.
The energy transfer cannot exceed a certain maximum, as seen earlier. This
maximum is for m2 >> me: 2meβ2γ2
2Z12e 4
dE 4π n e Z12e 4
m v 3γ 2
Z e2
=
ln e 12
one obtains bmin ≈ 1 2 so:
2
With: ∆ E = b 2 v 2m
me v1
Z1e ω
dx
γmv1
1
e
The result shows for low energies decreasing dE/dx for increasing E
and for higher energies an increase in dE/dx, due to the increase in γ
and v approaching c. The properties of the material enter via ne, i.e. only
the electron density matters.
Or: dE/dX = C1/v12 ln C2 v13γ2
dE/dx for pions as computed with Bethe-Bloch equation
dE/dx divided
by density ρ
(approximately
material
independent)
slope due to 1/v2
relativistic rise
due to ln γ
ρ about proportional to ne,
as ne = na Z = NA ρ Z / A, -> ne ≈ NA ρ / 2
high βγ:
dE/dx
independent
of βγ
due to
density
effect,
"Fermi
plateau"
From PDG, Summer 2002
Fig.2 Typical curves of dE/dX for pions as a function of their energy. The value drops for
increasing energy, and after reaching a minimum (‘minimum ionizing particle’ the value rises
slowly (‘relativistic rise’).
Calculation of distribution function of dE/dX:
Φ(W )dW dx = n a
dσ(W )
dW dx
dW
number of atoms
per unit of volume
Probability energy loss
between W and W+dW
∞
"Primary Ionization rate": q = n a
∫
0
dσ(W )
dW ; q dx is the probability for energy
dW
loss in dx
Define χ(W,x)dW as the probability for energy loss between W and W+dW
in a length x of absorber and then write down the "transport equation" for a
a length of dx of absorber at position x in the absorber:
x
dx
Number of particles with energy loss between W and W+dW in x+dx (I)
=
number of particles with energy loss between W and W+dW in x
(II)
number of particles with energy loss > 0 in dx
(III)
+
number of particles with energy loss between W-e and W-e+dW in x (IV)
and energy loss e in dx
I : Nχ(W,x + dx)dW
II : Nχ(W,x)dW
III : Nχ(W,x)dW qdx
∞
IV : N
∫ χ(W − ε,x)dWΦ(ε)dxdε
ε= 0
∞
Nχ(W,x + dx)dW − Nχ(W,x)dW = N
∫ χ(W − ε,x)dWΦ(ε)dxdε − Nχ(W,x)dW qdx
ε= 0
or:
∂χ(W,x) W
= ∫ Φ(ε)χ(W − ε,x)dε − qχ(W,x)
∂x
ε= 0
We skip the evaluation but present the results instead:
From PDG, Summer 2002
Fig. 3: typical Landau distributions. The tail can be associated with ‘head-on’ collisions.
Number
of
events
Not a simple
example of a
Landau
curve !
Part of electrons
stop in detector,
rest either loses
part of their
energy when
passing through
the detector or
backscatter out of
the detector or
create escaping
bremsstrahlung
Fig.4. Another example of dE/dX: electrons with energy of 1 MeV traverse through a Si
detector (layer thickness 0.53 mm). The curve has partly a Landau shape, but deviates
towards the highest energy. Some electrons are stopped, and thus absorbed in the Si layer.
This causes the clear peak in the right-hand side of the spectrum.
The measured spectrum may be compared with a MonteCarlo simulation in which the dE/dX
process is digitally processes.
The relativistic rise can be applied for particle identification:
Relativistic rise largest for gasses, can be used for measurement of β,
i.e. of velocity, and therefore for particle identification,
BUT
many independent measurements are needed for obtaining a reasonable
accuracy. This is possible with a Time Projection Chamber (TPC), to be
discussed later in the course. Usually energy losses corresponding to
the tail of the Landau distribution are not taken into account.
For relatively low energies βγ < ~ 2 particle identification using energy
loss measurement is easier.
Time Projection Chamber (TPC): 2D/3D Drift Chamber
The Ultimate Wire (drift) Chamber
track of
charged
particle
E-field
(and B-field)
Wire plane
Wire Plane
+
Readout Pads
Pad plane
Fig. 5: The Time Projection Chamber(TPC). The charge signals, measured on the (cathode)
strips, are proportional to dE/dX of the track passing the gaseous volume.
Particle identification in the relativistic rise region with the ALEPH TPC
http://alephwww.cern.ch/ALEPHGENERAL/reports/figures/detector/index.html
Fig.6: Particle identifications from dE/dX as a function of the particle’s momentum.
Bubble chamber photograph
shows different bubble
density along tracks for
different particle momenta
and particle type.
http://physics.hallym.ac.kr/education/hep/adventure/bubble_chamber.html
Fig. 7: bubble chamber picture showing clear dependence of dE/dX on the particle’s
momentum.
Bremstrahlung
Photon (invisible)
Kink in trajectory
Curvature smaller
after kink: due to lower
momentum
Hans Albrecht Bethe
e
Walter Heitler
http://teachers.web.cern.ch/teachers/archiv/HST2002/Bubblech/mbitu/electromag-events1.htm
This is another interaction between an energetic charged particle and material. The particle
interacts with a bounded charged particle in material, and as a result a gamma (photon) is
being created. The energy of the emitted photon and the (continuing) particle equals the
original energy op the incoming energetic particle.
The static particle in the material must be ‘bounded’ in order to transfer moment.
Bremstrahlung can NOT occur in vacuum, although conservation of energy would allow this.
[In the following:
1- note that the momentum ǁPǁ of a photon with energy E equals E/c. The speed of light c
often occurs as term in equasions and cancel: see chapter Accelerators.
2- An interaction or a process between particles is often represented by a Feynman
diagram, in which in- and outgoing particles are indicated.]
Photons can not be radiated in vacuum
Before: Ei2 - pi2 = me2
Ef,pf
e
E γ = pγ
γ
e
Ei,pi
Feynman diagram
2
2
2
After: (E f + E γ ) − (pf + pγ ) = me
2E f E γ = 2pf • pγ
p
E f = pf • γ
E f < p f not possible
pγ
Photons can be radiated in the field of an object with
charge
Ef,pf
e
Ei = Ef + Eγ
Very heavy pointlike
spinless object with
charge Z2
(atomic nucleus)
γ
e
Ei,pi
γ
e
e
This diagram also contributes
Classically: electric field
accelerates electron.
Quantum-mechanically: virtual photon
exchange decreases pf, so that
Ef2 - pf2 = me2 is satisfied
Z2
ESRF: European Synchrotron Radiation Facility, Grenoble, France
300 m circumference booster
synchrotron, 6 GeV
16 m linac, 200 MeV
Fig. 8: Application of Bremstrahlung. Accelerated electrons (6 GeV) are kept in orbit by
magnets with a vertical pointing B-field. While passing through this field, photons (mainly in
the keV range) are emitted.
The interaction of photons with material
The three main interactions are:
- photo-effect: the photon interacts with a electron which is in a bounded shell. The
electron is emitted with an energy equal to the photon energy minus the bounding
energy of the electron;
- Compton effect: the photon interacts with a (possibly free) charged particle. Some of
the energy of the photon is transferred to the charged particle. The photon (with
reduced energy) and charged particle are outcoming particles: total energy and
momentum are conserved.
- Pair production: the conversion of a photon into an electron-positron pair. This can not
occur in vacuum (violation of conservation laws: see Bremstrahlung: identical
Feynman diagram with inverted time and charge sign!).
From PDG, Summer 2002
Fig.9: Probability for photo-effect, Compton scattering and Pair Production with Lead, as a
function of the photon energy.
Cerenkov radiation
This interaction occurs if a charged particle tranverses a medium (material) in which the local
speed of light (= c/n: n is the refraction index of the medium) is SMALLER than the speed of
the charged particle. The effect is very much like a jet airplane flying faster than the speed of
sound: it drags a cone with ‘sonic boom’.
Fig. 10: The origine of Cerenkov radiation. UV ligh is emitted from a cone which opening
angle is determined by the refraction index of the medium and the speed of the particle.
Fig. 11: Cerenkov light from the core of a nuclear reactor immerged in a water basin.