interactions of particles with matter - department of physics...nov 2020 a. weber 5 bethe-bloch (1)...

Interaction of Particles

with Matter

Alfons Weber

STFC/RAL & University of

Oxford

Graduate Lecture 2020

Nov 2020 A. Weber 2

Table of Contents

Bethe-Bloch Formula

Energy loss of heavy particles by Ionisation

Multiple Scattering

Change of particle direction in Matter

Cerenkov Radiation

Light emitted by particles travelling in

dielectric materials

Transition Radiation

Light emitted on traversing matter boundary

Nov 2020 A. Weber 3

Nov 2020 A. Weber 4

Bethe-Bloch Formula

Describes how heavy particles (m>>me)

loose energy when travelling through

material

Exact theoretical treatment difficult

Atomic excitations

Screening

Bulk effects

Simplified derivation ala MPhys course

Phenomenological description

Nov 2020 A. Weber 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

br

θx

y

Nov 2020 A. Weber 6

Bethe-Bloch (2)

2

0

1

2x

Zzep dtF

c b

Force on projectile

Change of momentum of target/projectile

Energy transferred to target

2 23

2 2

0 0

cos cos4 4

x

Zze ZzeF

r b

2 2 2 4

2 2 2

0

1

2 2 (2 ) ( )

p Z z eE

M M c b

Nov 2020 A. Weber 7

Bethe-Bloch (3)

Consider α-particle scattering off Atom

Mass of nucleus: M=A*mp

Mass of electron: M=me

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 ZE

M M c b M

2 4

2 2 2 2

0

2 1( )

(4 )e

e

z eE b E

m c b

Nov 2020 A. Weber 8

Bethe-Bloch (4)

Energy transfer is determined by impact

parameter b

Integration over all impact parameters

bdb

ze

2 (number of electrons / unit area )

=2 A

dnb

db

Nb Z x

A

Nov 2020 A. Weber 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

2

0

dd ( ) 2 ln

d

ln

with 24

bbe

e b

b

Ee

E

A

e

m cn ZzE b E b C x b

b A

m c ZzC x E

A

eC N

m c

Nov 2020 A. Weber 10

Bethe-Bloch (6)

Simple approximations for

From relativistic kinematics

Inelastic collision

Results in the following expression

min 0 average ionisation energyE I

2 2 2 22

2

0

22 lne em c m cE ZzC

x A I

2 2 22 2 2

max 2

22

1 2

ee

e e

m cE m c

m m

M M


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 222max

2 2

0

21 ( )2 ln

2 2 2

e em c m c EE ZzC

x A I


Energy Loss Function

/ stopping powerE

x


Average Ionisation Energy


Density Correction

Density Correction does depend on

material

with

x = log10(p/M)

C, δ0, x0 material dependant constants


Different Materials (1)


Different Materials (2)


Particle Range/Stopping Power


Energy-loss in Tracking Chamber


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


Straggling (2)

Simple parameterisation

Landau function

Better to use Vavilov distribution

2

2

1 1( ) exp ( )

22

with e

f e

E E

m c ZzC x

A


Straggling (3)


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 Emax

δ-Ray


Restricted dE/dx

Some detector only measure energy loss

up to a certain upper limit Ecut

Truncated mean measurement

δ-rays leaving the detector

2 2 2 22

2 2

0

2

max

212 ln

2

( ) 1

2 2

cut

e e cut

E E

cut

m c m c EE ZzC

x A I

E

E


Electrons

Electrons are different light

Bremsstrahlung

Pair production


Table of Contents

Bethe-Bloch Formula


Multiple Scattering


Cerenkov Radiation






Multiple Scattering

Particles don’t only loose energy …

… they also change direction


MS Theory

Average scattering angle is roughly

Gaussian for small deflection angles

With

Angular distributions are given by

0

0 0

0

13.6 MeV1 0.038ln

radiation length

x xz

cp X X

X

2

2 2

0 0

2

2

00

1exp

2 4

1exp

22

space

plane

plane

dN

d

dN

d


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


Correlations (W. Allison)

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

Note that this is a Log-log-log plot - the cross section varies over 20

and more decades!

log kL

2

18

17

7

log kT

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 pL or

energy transfer

(16 decades)

Log pT transfer

(10 decades)

Log cross

section

(30

decades)


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

Transition radiation


Moving charge in dielectric medium

Wave front comes out at certain angle

Cherenkov Radiation

1cos c

n

slow fast


Cherenkov Radiation (2)

How many Cherenkov photons are

detected?2

2

2

2

2 2 2

0 2 2

( )sin ( )d

1( ) 1 d

11

with ( ) Efficiency to detect photons of energy

radiator length

electron radius

c

e e

e e

e

zN L E E E

r m c

zL E E

r m c n

LNn

E E

L

r


Different Cherenkov Detectors

Threshold Detectors

Yes/No on whether the speed is β>1/n

Differential Detectors

βmax > β > βmin

Ring-Imaging Detectors

Measure β


Threshold Counter

Particle travel through radiator

Cherenkov radiation


Differential Detectors

Will reflect light onto PMT for certain

angles only β Selection


Ring Imaging Detectors (1)



More clever geometries are possible

Two radiators One photon detector



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


22 vq

vacuummedium

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

11 vq

33 qv

Transition Radiation (2)


Transition Radiation (3)

Consider relativistic particle traversing a

boundary from material (1) to material (2)

Total energy radiated

Can be used to measure γ

22 2

2

2 2 2 2 2 2 2

d 1 1

d d / 1/ 1/

plasma frequency

p

p

N z


Transition Radiation Detector


ATLAS TRTracker

ATLAS

ExperimentInner Detector:

pixel, silicon and straw tubes

Combination of Central Tracker and

TR for electron identification


Atlas TRT (II)


Atlas TRT (III)

TRT senses

ionisation

transition radiation

only electron produce

TR in radiator

e± / π separationElectrons with radiator

Electrons without radiator

Bod -> J/yKo

s

High threshold hits


Table of Contents

Bethe-Bloch Formula


Multiple Scattering


Cerenkov Radiation



Transition radiation



Bibliography

This lecture https://www2.physics.ox.ac.uk/contacts/people/weber

PDG online: Experimental Methods https://pdg.lbl.gov/2020/reviews/contents_sports.html

Passage of particles through matter

Particle detectors …

References therein, especially Rossi

Lecture notes of Chris Booth, Sheffield http://cbooth.staff.shef.ac.uk/phy6040det/

Or just it!

https://www2.physics.ox.ac.uk/contacts/people/weber

https://pdg.lbl.gov/2020/reviews/contents_sports.html

http://cbooth.staff.shef.ac.uk/phy6040det/

interactions of particles with matter - department of physics...nov 2020 a. weber 5 bethe-bloch (1)...

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