Interaction of Particles with Matter

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Interaction of Particles with Matter

Alfons Weber

STFC & University of Oxford Graduate Lecture 2008

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Bethe-Bloch Formula

Describes how heavy particles (m>>m_e) loose energy when travelling through

material

Exact theoretical treatment difficult

Atomic excitations

Screening

Bulk effects

Simplified derivation ala MPhys course

(5)

Bethe-Bloch (1)

Consider particle of charge ze, passing a stationary charge Ze

Assume

Target is non-relativistic

Target does not move

Calculate

Momentum transfer

Energy transferred to target

ze

Ze r b

θ

x

y

(6)

Bethe-Bloch (2)

2

0

1

x 2 p dtF Zze

πε β c b

∞

∆ =

∫

−∞ =

Force on projectile

Change of momentum of target/projectile

Energy transferred

2 2

3

2 2

0 0

cos cos

4 4

x

Zze Zze

F r θ b θ

πε πε

= =

(7)

Bethe-Bloch (3)

Consider α-particle scattering off Atom

Mass of nucleus: M=A*m_p

Mass of electron: M=m_e

But energy transfer is

Energy transfer to single electron is

2 2 2 4 2

2 2 2

0

1

2 2 (2 ) ( )

p Z z e Z

E M M πε β c b M

∆ = ∆ = ∝

2 4

2 2 2 2

0

2 1

( ) (4 )

e

E b E z e

m c πε β b

= ∆ =

(8)

Bethe-Bloch (4)

Energy transfer is determined by impact parameter b

Integration over all impact parameters

b

db ze

2 (number of electrons / unit area ) dn = πb×

(9)

Bethe-Bloch (5)

Calculate average energy loss

There must be limits

material dependence is in the calculation of the limits

[ ] [ ]

max

max min min

max min

2 2

2

2 2

2

2 0

d d ( ) 2 ln

d

ln

with 2

4

b

e b

b

e E

E

A

e

m c

n Zz

E b E b C x b

b A

m c Zz

C x E

A C N e

m c

β ρ

π πε

∆ = = ∆

= ∆

 

=  

 

∫

(10)

Bethe-Bloch (6)

Simple approximations for

From relativistic kinematics

Inelastic collision

Results in the following expression

min 0 average ionisation energy E = ≡I

2 2 2 2 2

2

m c m c

E Zz ^

γ β

^

∆ ≈

2 2 2

max 2

2 2

1 2

e

e e

E m c m c

m m

M M

γ β γ β

γ

= ≈

 

+ +  

 

(11)

Bethe-Bloch (7)

This was just a simplified derivation

Incomplete

Just to get an idea how it is done

The (approximated) true answer is

with

ε screening correction of inner electrons

δ density correction (polarisation in medium)

2 2 2 2 2

max 2

2 2

0

2

1 ( )

2 ln

2 2 2

e e

m c m c E

E Zz

x C A I

γ β ε δ β

ρ β

β

   

∆ =    − − − 

∆    

(12)

Energy Loss Function

/ stopping power E

x ρ

∆ =

∆

(13)

Average Ionisation Energy

(14)

Density Correction

Density Correction does depend on material

with

x = log₁₀(p/M)

C, δ₀, x₀ material dependant constants

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Different Materials (1)

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Different Materials (2)

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Particle Range/Stopping Power

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Energy-loss in Tracking Chamber

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Straggling (1)

So far we have only discussed the mean energy loss

Actual energy loss will scatter around the mean value

Difficult to calculate

parameterization exist in GEANT and some standalone software libraries

From of distribution is important as energy loss distribution is often used for calibrating the detector

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Straggling (2)

Simple parameterisation

Landau function

2 2

1 1

( ) exp ( )

2 2

with

e

f e

E E

m c Zz

C x

A

λ λ λ

π

λ

β ρ

 − 

=  − + 

 

∆ − ∆

=

∆

(21)

Straggling (3)

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δ-Rays

Energy loss distribution is not Gaussian around mean.

In rare cases a lot of energy is transferred to a single electron

If one excludes δ-rays, the average energy loss changes

Equivalent of changing E

δ -Ray

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Restricted dE/dx

Some detector only measure energy loss up to a certain upper limit E_cut

Truncated mean measurement

δ-rays leaving the detector

2 2 2 2 2

2 2

0

2

max

2

2 1 ln

2

1 ( )

2 2

cut

e e cut

E E

cut

m c m c E

E Zz

x C A I

E E

ρ γ β β

ε δ β β

<

 ∆  =   

    

∆  

  

  

−  +  − − 

  

(24)

Electrons

Electrons are different light

Bremsstrahlung

Pair production

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Multiple Scattering

Particles don’t only loose energy …

(27)

MS Theory

Average scattering angle is roughly Gaussian for small deflection angles

With

Angular distributions are given by

0

0 0

0

13.6 MeV

1 0.038 ln radiation length

x x

cp z X X

X

θ β

  

=  +  

 

 

≡

2

2 2

0 0

2 2 0 0

1 exp

2 4

1 exp 2 2

space

plane plane

dN d dN d

θ

πθ θ

θ

θ πθ θ

 

∝  − 

Ω  

 

∝  − 

 

(28)

Correlations

Multiple scattering and dE/dx are normally treated to be independent from each

Not true

large scatter large energy transfer

small scatter small energy transfer

Detailed calculation is difficult, but possible

Wade Allison & John Cobb are the experts

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Correlations (W. Allison)

Example: Calculated cross section for 500MeV/c µ in Argon gas.

Note that this is a Log-log-log plot - the cross section varies over 20 and more decades!

log k_L

2

18

17

7

log k_T

whole atoms at

low Q² (dipole region)

electrons at high

Q²

electrons backwards in

CM nuclear small angle

scattering (suppressed by screening)

nuclear backward scattering in CM (suppressed by nuclear

form factor)

Log p_L or energy transfer

(16 decades)

Log p_T transfer (10 decades) Log cross

section (30 decades)

(30)

Signals from Particles in Matter

Signals in particle detectors are mainly due to ionisation

Gas chambers

Silicon detectors

Scintillators

Direct light emission by particles travelling faster than the speed of light in a medium

Cherenkov radiation

Similar, but not identical

(31)

Moving charge in dielectric medium

Wave front comes out at certain angle

Cherenkov Radiation

cos _c 1 θ n

= β

slow fast

(32)

Cherenkov Radiation (2)

How many Cherenkov photons are detected?

2

2 2

2

2 2 2

0 2 2

( ) sin ( )d

( ) 1 1 d

1 1

with ( ) Efficiency to detect photons of energy

c e e

e e

N L z E E E

r m c

L z E E

r m c n

LN n

E E

γ α ε θ

α ε

β

β ε

=

 

=  − 

 

 

 

≈  − 

 

=

∫

(33)

Different Cherenkov Detectors

Threshold Detectors

Yes/No on whether the speed is β>1/n

Differential Detectors

β_max> β > β_min

Ring-Imaging Detectors

Measure β

(34)

Threshold Counter

Particle travel through radiator

Cherenkov radiation

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Differential Detectors

Will reflect light onto PMT for certain angles only β Selection

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Ring Imaging Detectors (1)

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Ring Imaging Detectors (2)

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Ring Imaging Detectors (3)

More clever geometries are possible

Two radiators One photon detector

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Transition Radiation

Transition radiation is produced, when a relativistic particle traverses an

inhomogeneous medium

Boundary between different materials with different diffractive index n.

Strange effect

What is generating the radiation?

Accelerated charges

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2

2 v

q

vacuum medium

Before the charge crosses the surface,

apparent charge q₁ with apparent transverse vel v₁

After the charge crosses the surface,

apparent charges q₂ and q₃ with apparent transverse vel v₂ and v₃

1

1 v

q

3

3 q

v

Transition Radiation (2)

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Transition Radiation (3)

Consider relativistic particle traversing a boundary from material (1) to material (2)

Total energy radiated

Can be used to measure γ

2 2 2

2

2 2 2 2 2 2 2

d 1 1

d d / 1/ 1/

plasma frequency

p p

N z α φ

ω π ω ω ω φ γ φ γ

ω

 

= × − 

Ω  + + + 

=

(42)

Transition Radiation Detector

(43)

ATLAS TRTracker

ATLAS

Experiment Inner Detector:

pixel, silicon and straw tubes

Combination of Central Tracker and TR for electron identification

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Atlas TRT (II)

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Atlas TRT (III)

TRT senses

ionisation

transition radiation

only electron produce TR in radiator

e^± / π separation

Electrons with radiator

Electrons without radiator

B^o_d -> J/ψK^o_s

High threshold hits

(46)

Bibliography

This lecture

http://www-pnp.physics.ox.ac.uk/~weber/teaching

PDG 2008 (chapter 27 & 28) and references therein

Especially Rossi

Lecture notes of Chris Booth, Sheffield

http://www.shef.ac.uk/physics/teaching/phy311

R. Bock, Particle Detector Brief Book

http://rkb.home.cern.ch/rkb/PH14pp/node1.html

Or just it!

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Interaction of Particles with Matter