Interaction with matter...Formula needs to be modified for electrons Bethe-Bloch formula is not...

Katharina Müller, Autumn 14 1

Understanding/description of interaction between matter and high energetic particles/radiationimportant:

● Principles of detection of particles/radiation

● Limitation of detectorsEfficiencyPosition resolutionEnergy resolutionTime resolution

● Impact on biological systemsRadiation damageRadiation protectionRadiation therapy

● Literature K. Kleinknecht, Teilchendetektoren

C. Grupen, Teilchendetektoren (BI Wissenschaftsverlag)

J. Ferbel, Experimental Techniques in High Energy Physics

Particle data group, chapter 26 , http://pdg.lbl.gov/pdg.html

The particle detector brief book http://rkb.home.cern.ch/rkb/titleD.html

dEdX: http://www.srim.org/SRIM/SRIMPICS/SRIM-High%20Velocity.pdf

Interaction with matter

http://pdg.lbl.gov/pdg.html

http://rkb.home.cern.ch/rkb/titleD.html


Interaction with matterEnergy loss per distance (dE/dx)

many different interactions, dominating processes depend on energy and particle type

Charged particles: defined reach

● Ionisation and excitation of electrons on shell→ Bethe-Bloch formula

● Coulomb Scattering: Scattering in Coulomb field of nucleussmall energy loss, but deflection

● Bremsstrahlung : dominant for low masses → radiation length

Photons: Absorption in matter, highly energy dependent – attenuation, no defined reach● Photo electrical effect● Compton scattering● Pair production

Hadrons: inelastic scattering ● Hadron+nucleus → π±,0 , K, p,n, fragments of nucleus

Z, A,

z, m, E, Θ

e± ,μ±, ± , , p,n,


Energy loss of charged particles in matterEnergy loss due to ionisation and excitation

Energy loss per distance (dE/dx) : many theoretical works

● N. Bohr Classical derivation

● Bethe, Bloch Quantum mechanical description

● L. Landau Charge distribution function

● E. Fermi Density correction

...


Ionisation: classical derivation (Bohr)Energy loss via inelastic collisions with electrons on shell → Ionisation and excitation

Assumptions: M(particle) >> me

Shell electron at rest (collision time small wrt orbital time)Projectile M, v = βc , charge ze, Target: charge Ze

Energy transfer onto target with mass mt: ΔE=Δp/(2m

t)

Momentum transfer: time integral

longitudinal forces cancel

only transversal forces important

p=∫−∞

∞

FCoulomb dt FC=1

40zZe2

/r2

F t=FC⋅b∣r∣

=FC⋅cos

F l x =−F l−x

Z e

r b

x

4

b: collision parameter

ze


Ionisation: classical derivation (Bohr) I

Transversal forces on projectile

Momentum transfer onto target

Integration →

Energy transfer onto target with mass mt

Now: Sum over all shell electrons Integration over all collision parameters b

p b=∫−∞

∞

FC t dt=∫−∞

∞

FC tdxv=

140

z Z e2

b2 ∫−∞

∞

cos3dx

b d /cos2

v

FC t=1

40

z Z e2

r2˙cos=

140

z Z e2

b2˙cos3

E b= p2

2 mt

=2 z2 e4

me c2

2 b21

40

≡1

shell electron Z = 1, mt=m

e

1/m: collisions with nucleus may be neglected

=1

40

z Z e2

b v ∫−/2

/2

cosd =1

40

2 z Z e2

b v

r=b/cosθ


Ionisation: classical derivation (Bohr) IIEnergy loss due to collisions with electrons in tube (length dx, thickness db) at radius b:

# of electrons: 2 π b db dx Ne electron density Ne = NA Z/A ρ

(NA Avogadro's constant, A mass number)

Integration over collision parameter from bmin to bmax

maximal energy transfer: central collision

minimal energy transfer: mean excitation energy I (averaged excitation potential per electron in the target)

−dE b= E bN e dV=4 z2 e4

me c2

2N e

dbb

dx

−dEdx

b=4 z2 e4

me c2

2N A

ZA

1b

db −dEdx

=4 z2 e4

me c2

2 N A

ZA

lnbmax

bmin

E b=22 me 2 c2

=2 z2 e4

me c2 2 b2

bmin=z e2

me c2

2

bmax=z e2

c 2

me I

b

db

dx

note: 1/β2


(dE/dx) classical derivation (Bohr) III

Classical formula for energy loss due to ionisation

Unit [dE/dx] = MeV /cm

Warning: usually energy loss is divided by density: [dE/dX] = MeV cm2/g

→ dE/dX almost independent of material only material dependence through Z/A and ln(1/I)

−dEdx

=4 z2 e4

me c2

2N A

ZA

lnbmax

bmin

bmin=z e2

me c2

2bmax=

z e2

c 2me I

−dEdx

=K z2

2

ZA

12

ln2c2

2

2 me

IK=

4 e4

c2 me

N A=0.31 MeV cm2/g


(dE/dx) Bethe Bloch Formula

full quantum mechanical derivation: Bethe-Bloch Formula

I mean excitation energyZ atomic numberA mass number of absorberValidity: 6 MeV < E < 6 GeV (π), generally 0.02< β<0.99 (1% accuracy!!)

T max: max kinematic energy transferred

z: Charge of particle dE/dX ∝ z2

Corrections

● δ(β) Density correction due to polarisation, important for high energies● C/Z Correction close to shell boundaries, relevant for small energies 1% @ βγ =0.3

−dEdX

=K z2 ZA

1

2[12

ln2 me c2

2

2 T max

I 2−

2−

2−

CZ]

K=4 e4

c2 me

N A=0.31 MeV cm2/g


(dE/dx) Bethe-Bloch-(Sternheimer) Formulaclassical result:

full quantum mechanical derivation: Bethe-Bloch Formula

describes mean energy loss or stopping power

●T max: max kinematic energy transferred from particle (M) onto electron

● small energies and particles with high masses: result equals classical result except factor 2 (imperfect description of very far collisions : binding of electrons cannot be neglected)

● independent of mass of particle ! (γm e<<M)● Unit MeV/(g/cm2) dx = ρ ds is material occupancy● dE/dX almost independent of target material● Formula needs to be modified for electrons ● Bethe-Bloch formula is not valid for slow particles βγ<0.02 where dE/dX ∝ β

−dEdX

=K z2

2

ZA

12

ln2 me c2 2 2

I

−dEdX

=K z2 ZA

1β2

[12

ln2 me c2β2γ2T max

I 2−β2−δ

2−

CZ]

T max=2 me c222

12 me/Mme /M 2

≈ 2 me c2

2

2me≪M

9

≈ M c2∞


Energy loss dE/dX

Titel

dE/dx of muons in copper

~β

~1/β2

~2 ln(γβ)

● βγ < 3 dE/dX ∝ 1/β2

● βγ ≃ 3.5 minimum dE/dX 1-1.7 MeV cm2/g

● βγ > 3.5 logarithmic rise dE/dX ≃ 2 MeV cm2/g● βγ > 1000 Bremsstrahlung

dominant● very low energy: BB not valid,

empirical models● low energies: shell corrections● high energies: density corr.

Minimum: βγ 3-4 minimum ionising particle (MIP) looses ~ 1-1.7 MeV/(g/cm2)

PDG http://pdg.lbl.gov/pdg.html



Shell correction−dEdX

=K z2 ZA

1

2[12

ln2 me c2

2

2 T max

I 2−

2−

2−

CZ]

● Shell corrections (C/Z) constitute a correction to slow particles (e.g for protons in the energy range of 1-100 MeV)

● Correction for the assumption that particle velo-city >> bound electron velocity

● Assumption that electron is at rest not valid

● Maximal about 6%

● Calculation with various approximations

● J. F. Ziegler, Applied Physics Reviews / J. Applied Phys-ics, 85, 1249-1272 (1999).

● http://www.srim.org/SRIM/SRIMPICS/IONIZ.htm

Copper: about 1% at βγ=0.3 (6 MeV Pion), decreases rapidly with velocity


Mean excitation energy I−dEdX

=K z2 ZA

1

2[12

ln2 me c2

2

2 T max

I 2−

2−

2−

CZ]

I characteristic for target material I = 16 Z 0.9 eV for Z > 1

I=10 eV·Z for Z>20

Example Argon: Z = 18, I = 215 eV, measured 190.8 eV (Ionisation energy 15.7eV)

Mean excitation energy > ionisation E Theoretical calculation of I has a long history. Summaries can be found in several reviews :U. Fano, Chr., Studies in Penetration of Charged Particles in Matter, Nucl. Sci. Rpt.. 39, U. S. National Academy of Sciences, Washington, 1-338 (1964). J. F. Ziegler, “Handbook of Stopping Cross-Sections for Energetic Ions in All Elements”, Pergamon Press (1980).




Charge dependence−dEdX

=K z2 ZA

1

2[12

ln2 me c2

2

2 T max

I 2−

2−

2−

CZ]

Tracks of ions in emulsion

Iron Z = 26 Thorium Z = 90

Energy loss depends quadraticallyon charge of projectilewidth allows estimate of charge


dE/dX discussion

● depends on velocity not on mass → particle identification

● βγ small dE/dx ∝ 1/β2

● βγ ≅ 3.5 broad minimum light absorbers: Z/A ≃ 0.5 ● dE/dx(min) ≃ 1.5 MeV /(g/cm2)

→ minimal ionising particles (MIP)

● βγ > 4 dE/dx ∝ 2 ln(βγ2) logarithmic (relativistic) rise

1/2

MIP

≅ 3-4

without

T max≈2 me c2 22

relativistic rise

−dEdX

=K z2 ZA

1

2[12

ln2 me c2

2

2 T max

I 2−

2−

2−

CZ]


dE/dxmin different materials

dE/dX depends on A, Z of target material

βγ≅ 3.5 broad minimum → minimum ionising particles (MIP)

H2 Z/A≅ 1 dE/dXmin ≅ 4 MeV /(g/cm2)others Z/A≅ 0.5 dE/dXmin ≅ 2 MeV /(g/cm2)

dE/dXmin ≅ 1-1.7 MeV /(g/cm2) only weak material dependence


−dEdX

=K z2 ZA

1

2[12

ln2 me c2

2

2 T max

I 2−

2−

2−

CZ]



dE/dX at minimumβγ → 3.5 broad minimum dE/dXmin ≅ 1-1.7 MeV/(g/cm2)only small material dependence

dE/dX(min) for different chemical elements fitted by a straight line Z>6




Relativistic RiseLorentz-contraction of field linestransversal component of E-field grows with γ → larger collision distances, more collisions

Saturation for high energies→ Fermi plateau Tmax

= E

Solids (dE/dX)β→1≈ 1.05-1.1 (dE/dX)MIP

Gases (dE/dX)β→1≈ 1.5 (dE/dX)MIP

δ-correction, density effect:E-field gets partially shielded for high densities due to polar-isation δ(γ) = 2 ln γ +ζ (ζ material constant)→ less collisions with far distant electrons, smaller dE/dX

Fermi plateauwithout correction

with correction

at rest

E- field


Particle identification

dE/dx only depends on velocity not on mass of particle → Particle identification

−dEdx

≈K z2 ZA

1

2[ ln

2 me c2

2

2

I−

2−

2−

CZ] as long as 2me≪M

ALICE TPC measured energy loss

• Heavy particles dE/dx well described by Bethe-Bloch formula

• Electrons not described by Bethe-Bloch

http://aliceinfo.cern.ch/


ALICE TPC

88 m3 cylinder filled with gas, 5.1 m long

Divided in two drift regions by the central electrode located at its axial centre.

Uniform electric field along the z-axis electrons drift towards the end plates

Signal amplification:avalanche effect near anode wires

Readout: 570132 pads in cathode of multi-wire proportional chamber

http://aliceinfo.cern.ch/Public/en/Chapter2/Chap2_TPC.html Cosmic muon in TPC


Restricted Bethe BlochParticle detectors don't measure energy loss, but energy deposited in detector.

Eg. highly energetic electrons may leave the detector → measured energy is too small

Restricted Bethe Bloch: Cut off parameter Tcut

:dE with T>Tcut

are neglected

Tupper

= min(Tcut

,Tmax

)

−dEdX

=K z2 ZA

1

2[12

ln2 me c2

2

2 T upper

I 2−

21

T upper

T cut

−

2−

CZ]


Range in mattermean range R/M (M mass of particle)

Formally: integration of Bethe-Bloch formula

statistical process → mean penetration depthrange in cm: divided by density!

Example 1 GeV particle in leadK+ M= 493.6 MeV βγ = 2.02 → R/M = 800 g cm-2 GeV-1

R = 395 g cm-2

ds = R= 35 cm in lead

Myon R/M = 7000 g cm-2 GeV-1 ds = 64 cm

Proton R/m = 220 g cm-2 GeV-1 ds = 18 cm

R=∫E

0dXdE

dE




Ionisation tracksEnergy loss largest for low energy particles → increase of energy deposits (ionisation) towards end of range (Bragg peak)

δ-(knock-on) electrons: have high enough energy for ionisation track

Tracks of mono energetic α particles in cloud cham-ber: range 8.6 cm, small variation


Range: Bragg curveBragg curve: describes energy loss of particle as function of penetration depth

Particle in matter → particle de-accelerates→ energy loss increases as particle gets slower

Bragg-Peak: largest energy deposit at end of the trackimportant for radiation therapy!

Penetration depth

70 MeV protons in water

Charged particles

Photons

Bragg peak


Medicine: radiation therapy

Possible to precisely deposit dose at well defined depth by varying beam energy

Application in medicine:Concentration of energy deposit at end of reach allows treatment of tumours with moderate exposure dose for the surrounding tissue, contrary to x-ray

Ionisation profile for 12C ions in water


Proton therapy at PSI

Spread out Bragg peaks (SOBP)Different absorbers → almost constant dose in region of tumour


Energy loss (Straggling)

Energy loss is statistical process N: number of collisions● ionisation loss distributed statistically● collisions with small dE more probable● large dE rare → electrons with keV (δ-electrons)● δ-rays have enough energy for ionisation● asymmetry: mean energy loss > most probable energy loss

Parametrisation: asymmetric Landau distribution

Landau-Tail: rare interactions with large energy transfer Tails up to the kinematic maximum

Bethe-Bloch gives mean energy loss

Thick layers or dense material: Gaussian distr.

dEdX: most probable value ≠mean energy loss

Δ E=∑i=0

Nδ E i

En

erg

y loss

pro

bab

ility

Energy loss


Landau distributionMeasurement

ΔEmp =82 keV ΔEmp = 56.5 keV

Landau: Probability (Δ E) =

ξ: material constant mean energy loss in layer x ξ= K ρ x /β2 depends on density, thickness and velocityΔ E mean E- loss, Δ Emp most probable E-loss

12

exp−12e−

= E− E mp/


δ Electrons

Energy distribution of secondary electrons:

(Spin 0)

d N 2

dT d x=12k z 2

ZA1

2F T

T 2

Example: 500 MeV pion in 300 μm Si: 5% produce an electron with T>166 keVimportant background in Cherenkov counterNumber of δ-electrons proportional z2/β2

F T =1−2T /T max


δ Electrons

Number of δ electrons proportional to z2/β2

1969 Mc Cusker: Evidence of quarks in air shower cores (Phys Rev Lett)Narrow tracks in cloud chambers: low ionisation→ charge < 1!


Summary dE/dx through ionisationdescribed by Bethe-Bloch Formula

● only small material dependence

● Energy loss ∝ z2 (particle)

● independent of mass → particle identification

● most of the energy is deposited towards the end of the range

● βγ < 3 dE/dX ∝ 1/β2

● βγ ≃ 3.5 minimum dE/dX 1-1.7 MeV cm2/g

● βγ > 3.5 logarithmic rise dE/dX ≃ 2 MeV cm2/g

● βγ > 1000 Bremsstrahlung dominant

● Bethe-Bloch is not valid for slow particles (βγ<0.05) (dE/dX ∝ β) here only phenomenological models available


Additional Material: Ranges

Range of electrons, protons and α particles in air and water


Additional Material: Ionisation yield

Material Density g/cm3 Eion

[eV] I[eV] W[eV] np [cm-1] n

t [cm-1]

H2

8.99 10-5 15.4 19.2 37 5.2 9.2He 1.78 10-4 24.5 41.8 41 5.9 7.8N

21.25 10-3 15.6 82 35 10 56

Ne 9.00 10-4 21.6 137 36 12 39Ar 1.78 10-3 15.7 188 26 29 94Xe 5.89 10-3 12.1 482 22 44 307C4H

102.67 10-3 10.8 48.3 23 46 195

Ar-f 1.4 15.4 188 29.6 98000Xe-f 3.06 12.1 482 15.6 245000Si 2.33 3.6 172 1000000

● W = E/nt mean energy per electron-ion pair

● np Primary ionisation: directly produced electron-ion pairs ~ 1.45 Z

● nT= n

P +secondary ionisation: add. electron-ion pairs through primary electrons

● W: mean energy for production of electron-ion pair● W higher than ionisation energy

● Semiconductors, W small, many electron-ion pairs, good resolution, small detectors


Additional material: dE/dX: Mixtures

Since BB applies to pure elements, direct measurements are needed in compounds for accurate values.Good approximation: weighted sum of loss rates of constituentsweighted according to fraction a

i of electrons

1dEd x

=∑ w i

idEd x

i

w i=a i A iA e ff

Z e ff=∑ a i Z iA e ff=∑ a i A i

ai is the molar fraction of element i with atomic weight A

i,

Aeff

is the atomic weight of all constituents

Tables for density and shell correction

Interaction with matter...Formula needs to be modified for electrons Bethe-Bloch formula is not...

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