Interaction of charged particles in matter

today electrons

Introductory remarks

• Electrons ‚feel‘ electro-magnetic and weak interaction

• Electrons couple to photons and W- and Z-Boson of weak interaction

• Electrons ‚talk‘ easily to photons, remember pair creation and

Bremsstrahlung

• Electron and photon detection are entangled, e.g. electro-magnetic

shower detection at high energies

Relevance of electrons in nuclear and particle physics some examples:

- beta decays

- conversion electrons (complementary to g-spectroscopy)

- electron beams for electron-nucleon-scattering (structure functions)

- electron-positron collisions at high energies provide clean source for

particle production and new discoveries

- electron is most abundant reaction product, often source of unwanted

and tremendous back ground

2222)/( mcere

Consider moving particle of charge ze passing by with v the stationary charge Zeze

vb

Ze

r q

y

Assumptions:

• momentum approximation: short inter-

action, only transversal moment. transfer

• target remains non-relativistic

Interaction of heavy charged

particles in matter

Bethe-Bloch

energies)(low correction shell :

energies)ic relativist (at correctiondensity : terms correction two

collision single in transferenergy max. : matter abs. of massatomic :

potential ionisation averaged :I matter abs. of number charge :

v/c : mass electron :

particle incoming of charge : cm10 x 2.81 radius electron class. :

matter absorbing ofdensity : mol10 x 6.022 constant Avogadro

13-

1- 23

C

WA

Z

m

zr

N

Z

C

I

Wvmz

A

ZcmrN

dx

dE

e

e

a

eeea

gg

g

max

2

2

2

max

22

2

222

1/1,

:

222

ln2

Energy loss of electrons and positrons via collisions with electrons on atoms and

electro-magnetic radiation (Bremsstrahlung)

collradtot dx

dE

dx

dE

dx

dE

Modified calculation of energy loss

- large scattering angle of incoming particle

- scattering of identical particles -> Q.M. interference

positrons electrons

of units inenergy kinetic

...)23(12

2ln2)(...1)(:)(

:

2)()/(2

)2(ln

12

22

2

22

2

2

22

FFF

cm

Z

CF

cmIA

ZcmrN

dx

dE

e

e

eea

coll

Collisional energy loss:

Interaction of electrons and positrons

Bremsstrahlung loss and ionisation loss for electrons and protons in copper

electron

2-4 MeV proton

> 3 GeV

Comparison of interactions

crit. energy

24 MeV

Variation in range and energy loss caused by statistics of absorbing collisions.

Energy loss distribution

- average energy loss –(dE/dx)

- energy loss straggling

- range straggling

heavy charged particle electrons

Range and energy loss

Electron range fluctuates considerably due to

multiple scattering.

Large energy transfer via collisions is possible.

Range distribution is smeared out .

Transmission of electrons in aluminium [1 g/cm2 = 3.7 mm]

Electron range

Polyethylene [1 g/cm2 = 9.0 mm]

Aluminium [1 g/cm2 = 3.7 mm]

Lead [1 g/cm2 = 0.88 mm]

Range variation below critical energy is

very energy dependent.

Electron range: low energies

– decay of 185W

185W -> 185Re + - + n

Q = 433 keV, T1/2 = 75.1 d

Neutrinos cause continuous electron

spectrum

Folding of electron energy spectrum and

range in matter shows exponential

attenuation

I = I0 exp(-mx)

– attenuation coefficient m

Exponential attenuation is not valid in

general, only for simple –decay

scheme.

Absorption of electrons from185W - decay

Absorption of -electron

Scattering cause large deflection, even

Backward angles and small energy losses.

Depending on incident angle!

Electrons leave matter in most cases even

below the detection threshold.

Reduced detection efficiency!

Strongly dependent on:

-Electron energy

- at low energies highest impact

- charge number Z of absorbing matter

Electron back scattering

h is backscattering coefficient, given for perpendicular incidence

- Feynman graphs

- Weizsäcker-Williams approximation: Bremsstrahlung

Bremsstrahlung

Electrons and positrons are particles with considerable energy loss via

Bremsstrahlung for electron energies higher than several tens of MeV.

2222)/( mcere m mass: mm=106 MeV electron mass: me=511 keV

Differential Bremsstrahlung cross section

nuclei of fieldin electrons ofn interactio ,correction Coulomb :)(

100 :Parameter screening )(),( :function screening

energyelectron initial incoming:E

radiation after energy electron :E E/E

137

1

)(ln3

1

4

)(

3

2)(ln

3

1

4

)()1(4

3/1

0

2

21

0

0

21222

Zf

EZE

hcm

ZfZZfZd

rZd

e

e

n

n

n

E0

E

Bremsstrahlung

Screening of nuclear charge depends on

(i) initial energy z~ 12-25 for 1 MeV electrons, no screening

z~ 0,1-0,3 for 100 MeV electrons, complete screening

(ii) energy of Bremsstrahlung photon vs. remaining energy varies from small numbers for soft

photons to highest numbers for head on collisions causing high energy radiation

Differential Bremsstrahlung cross section

EZE

hcm

EZE

hcm

h

ZfZZfZd

rZd

ee

e

3/1

0

2

3/1

0

2

0

0

21222

100100 :Parameter screeningImportant

energy photon :

radiation after energy electron :E

energyelectron initial incoming:E

E/E

)(ln3

1

4

)(

3

2)(ln

3

1

4

)()1(4

nn

n

n

n

E0

E

Bremsstrahlung

nnn

n

n

nnn

n

n

n

n

n

n

n

n

dEd

dh

E

h

ENEdE

d

dhN

dx

dE

ZfZd

rZd

Zfhcm

EEdrZd

rad

rad

rad

e

e

e

)(1

and A

NN )(

N energy)photon (xsection integral from lossenergy Radiativ

9)()183ln()

3

21(4

:screening complete 0For

)(2

12ln)

3

21(4

:screening no 1For

,0

00

00

a0,0

0

3/1222

2

0222

0

0

N: number of atoms/cm3

Electron interaction: Bremsstrahlung

Two solutions for limiting extrem cases:Low energies

high energies

)(18

1)183ln(4

0,137

)(3

12ln4

1,137

3/122

3/12

2

022

3/122

ZfZrZ

Zcm

Zfcm

ErZ

Zcmcm

erad

e

e

erad

ee

:screening complete E For

:screening no For

0

High energy limit is independent of electron energy E0

Bremsstrahlung

)(

18

1)183ln(4 3/122 ZfZrZ erad

Collisions vs. Bremsstrahlung

Compare collision or ionisation vs Bremsstrahlung:

- ionisation increases logarithmic in E and linear in Z

- bremsstrahlung increases linear in E and quadratic in Z

- ionisation: quasi-continuous energy loss

- bremsstrahlung: emission of one or two photons can give away

full electron energy -> huge fluctuations

Z

CF

cmIA

ZcmrN

dx

dE

e

eea

coll

2)()/(2

)2(ln

12

22

2

2

22

)(18

1)183ln(4)(

1

)(

N energy)photon (xsection integral from lossenergy Radiativ

3/122

,0

00

0,0

0

0

0

ZfZrZdEd

dh

E

NEdEd

dhN

dx

dE

erad

rad

rad

nnn

n

nnn

n

n

n

Critical energy Ec:

2.1

800

Z

MeVEEE

dx

dE

dx

dEcc

collrad

for

Typical values:

material critical energy Ec

Pb 9.51 MeV

Al 51.0 MeV

Fe 27.4 MeV

Cu 24.8 MeV

Luft 102 MeV

NaI 17.4 MeV

H2O 92 MeV

Radiation loss

radiation lengthDistance in matter related to a radiative energy loss by 1/e

)(183ln)1(41

length radiation is 1/ andmatter in distance is

)exp(

DEQ ofSolution

energy fromt independen is (ii)

ignored loss coll. (i)

: 0)(limit energtichigh

/

3/12

0

ZfZrA

NZZ

L

NLx

L

xEE

dxNEdE

ea

rad

radrad

rad

rad

rad

Radiation length

Radiation length

Typical values:

Material cm

Pb 0.56

Al 8.90

Fe 1.76

Cu 1.43

NaI 2.59

BGO 1.12

Org.Scint. 42.40

H2O 36.10

Air 30050.00

Radiation length

collradtot dx

dE

dx

dE

dx

dE

positron ...)23(12

2ln2)( electron ...1)( :)(

of unitsin energy Kin. :

2)()/(2

)2(ln

12

22

2

22

2

2

22

FFF

cm

Z

CF

cmIA

ZcmrN

dx

dE

e

e

eea

coll

Energy loss via collisions with electrons:

Kritical energy Ec:

2.1

800 for

Z

MeVEEE

dx

dE

dx

dEcc

collrad

Summary: interactions of electrons

radiation lengthRadiational energy loss by 1/e

summary

Compare collision or ionisation vs Bremsstrahlung:

- Ionisation increases logarithmic in E and linear in Z

- Bremsstrahlung increases linear in E and quadratic in Z

- Ionisation: quasi-continuos energy loss

- Bremsstrahlung: emission of one or two photons can give away

full electron energy -> huge fluctuations

Z

CF

cmIA

ZcmrN

dx

dE

e

eea

coll

2)()/(2

)2(ln

12

22

2

2

22

)(18

1)183ln(4)(

1

)(

N energy)photon (xsection integral from lossenergy Radiativ

3/122

,0

00

0,0

0

0

0

ZfZrZdEd

dh

E

NEdEd

dhN

dx

dE

erad

rad

rad

nnn

n

nnn

n

n

n