Interaction of Charged Particles with Matter

W. Udo Schröder, 2009

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Interaction of Charged Particles with

Matter

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Literature/Tutorials

James Ziegler2


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Main Interactions of Charged Particles

Dominant type of interaction: inelastic collisions with electrons

Atomic excitation ionization fluorescence phosphorescence

Less important (CP) collisions with nuclei :

2 2 26 22 10 6

2 16 210 10 10

10nucl nucl

atom Z

R Z cm Za cm

Most interactions of charged particles with material components are collisions with atomic electrons.Nuclear collisions are noticeable only at low kinetic energies.

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Stochastic Multiple Scattering and Straggling

U ions, 60 keV, straggling in range and angle Target of layers (“absorbers” Be, Au, Si)

Prob

abilit

y

Range (Å)

Range is diffuse: “Range straggling”

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Phenomenological Model of Energy Loss in MatterBethe et al. (1930-1953), Lindhardt’s electron theory Describes energy loss through ionization, incoming ions are fully stripped

Estimate of trends/Order of magnitude E=particle kinetic energy, e- at rest

2p

e Coul coll 2

22 e

ee

4 22p

e2e

e Z 2rp F tvr

pd E( r , x ) 2 r dr dx N

2m

4 e Zd E( r , x ) 1Ndr dx rm v

r

dx

electron

shell

dr

Zp,v

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Phenomenological Model of Energy Loss in MatterIntegrate over radial coordinate:

r 2max

rmi

min

n4 2

e2e

axp m

dE( x) d E( r , x )drdx dr dx

4 e ZdE( x) N lndx v r

rm

Estimate of radial limits: Ne = e- density le = electronic wave length, IE=ionization energy

2e e e

2 2coll e

mi

x

e

n

m ea

h p h 1 m v

t v T 1 1 1

non adiabati

IE / h average e orbital f

c motion

requ y

r

r

enc

l

r

dx

electron

shell

dr

Zp,v

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Phenomenological Model of Energy Loss in Matter

Further: 4 2 2

p ee2 2

e e

4 e Z 2m vdE N lndx m v h 1

Insert: r = number density of atoms (#atoms/volume),

ZT = atomic number of target Ne = ZT · r

T Te 0.19

T T T

(12 Z 7 ) eV for Z 13IE h

(9.76 Z 58.8 Z ) eV for Z 13

Remember: This is only a rough estimate! Calibrate real detectors

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4 2 2

p eT2 2

e

e Z 2m v1 dE 4 Z lndx m v IE 1

r


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Bethe-Bloch Equation

Quantum mechanical calculation (for heavy particles M»me):

r = matter densityZT = atomic number of targetAT = mass number of target

T Te 0.19

T T T

(12 Z 7 ) eV for Z 13IE h

(9.76 Z 58.8 Z ) eV for Z 13

dx

(Zp, Ap, E)

E-dE

ZT AT r

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22 2 2p 2eT2 22

T

Z 2m cZ1 dE MeV0.1535 ln 2dx A g cmIE 1

r


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Minimum Ionizing Particles

Specific Energy Loss of Protons in Silicon

2 2p 2 2eT

2 2 2T

Z 2m cZ1dE 2 2 MeV0.1535 ln ln 1 2dx A IE g cm

r

Bethe-Bloch Formula

At high (relativistic energies, the terms become dominant. In addition, radiation losses (bremsstrahlung) and r-dependent plasma effects become important.dE/dx(E) has

minimum“minimum-ionizing

particles (mip)”examples: e±, m±

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Theoretical E-Loss Curves

Mg

He

Units MeV/Nucleon

Loss:MeV/A per mg/cm2

Bragg maximum

Semi-quantitative only: Expt values smaller both at small and large energies recharging effects for projectile

MathCad Program


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Theory and Practice for Heavy Ions

Theoretical energy loss in material of finite thickness is obtained from integration of the Bethe-Bloch formula or equivalent.Actual data may differ: Calibration required

Energy lost by various ions in a 15.9 mm Si transmission detector vs. ion energy per nucleon (mass number A). Curves represent different theories.


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Range and Stopping Power

Path length of trajectory

1

0

E

s sdE'R(E ) ds cos dE ' cosds

Range

S(E ) ds R(E )

Scattering angle s path variable sStochastic multiple scattering process produces straggling in range, energy loss, angle

dEds

Stopping power

Prob

abilit

y

Range (Å)

s

sds


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Range and Specific Ionization

Stopping power dE/dx (specific energy loss) depends on energy E and therefore on x

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# e ion pairsdEdx unit length

particles :1 dE 7 10 pairsIE dx mm

Bragg Curve

p

6600 ip/mm

2750 ip/mm

Highest E loss close to end of path Bragg maximum

Main E-loss mechanism: ionization, production of d electrons, electron-ion pairs

E-loss in Air: 1atm, 150C


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Stopping Power Isotopic Scaling Laws

2p 2

pp

2p

p

1p2p

d m v 2dE EZ fdx dx A

Zdv g vdx A

AdvR(v) dv h vdx Z

p2p

A0.56

Z

p2p

A0.12

Z

R(Be)/R(Ar)=2.97 expt 4.67 theoZeff ŧ Zp effective charge

Describes well the difference of R for different isotopes of a given element, but:


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Interactions of Neutrons via Secondaries

Characteristic secondary nuclear radiation/products: 1.charged particles (n,p), (n, ),…2.neutrons (n,n’), (n,2n’),…3.fission fragments (n,f) 4. g-rays (n, g)

No electric charge no direct atomic ionization only collisions and reactions with nuclei 10-6 x weaker absorption than charged particlesProcesses depend on available n energy En:En ~ 1/40 eV (= kBT) Slow diffusion, capture by nucleiEn < 10 MeV Elastic scattering, capture, nucl. excitationEn > 10 MeV Elastic+inel. scattering, various nuclear

reactions, secondary charged reaction products


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Neutron Cross Sections

1b=10-24cm2=100fm2


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Neutron Mean Free Path

FWHM 2.35

xN x N 0 e1 1 mfp

l

lm r

r: atomic density : cross sectionl = average path length in medium between 2 collisions

Mean Free Path of Neutrons in Water

Data: Neutron-mfp-Water


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Neutron mean free path data (H2O): http://t2.lanl.gov/data/ E_n [MeV] l [cm] E_n [MeV] l [cm] E_n [MeV] l [cm] E_n [MeV] l [cm] E_n [MeV] l [cm]

0.1000 1.0503 2.1000 4.5418 4.1000 6.7341 6.1000 8.6268 8.1000 10.7469

0.2000 1.3682 2.2000 4.7218 4.2000 6.3284 6.2000 9.0019 8.2000 10.3833

0.3000 1.6434 2.3000 5.0593 4.3000 6.3865 6.3000 9.1267 8.3000 10.3864

0.4000 1.7509 2.4000 5.2179 4.4000 6.5267 6.4000 8.5441 8.4000 10.6130

0.5000 2.0080 2.5000 5.1503 4.5000 7.0251 6.5000 9.6126 8.5000 10.8536

0.6000 2.2918 2.6000 5.2246 4.6000 7.1931 6.6000 9.3369 8.6000 10.9526

0.7000 2.4905 2.7000 5.3192 4.7000 7.4256 6.7000 9.3898 8.7000 10.7028

0.8000 2.6619 2.8000 5.4220 4.8000 7.4753 6.8000 9.0214 8.8000 10.7549

0.9000 2.7671 2.9000 5.5134 4.9000 7.4738 6.9000 9.7500 8.9000 11.2191

1.0000 2.4332 3.0000 5.5816 5.0000 7.7694 7.0000 9.7829 9.0000 11.2089

1.1000 2.9994 3.1000 5.5637 5.1000 7.1211 7.1000 9.7458 9.1000 11.2229

1.2000 3.2368 3.2000 5.3438 5.2000 7.7259 7.2000 8.9141 9.2000 11.5291

1.3000 2.9841 3.3000 5.1413 5.3000 8.0052 7.3000 9.5904 9.3000 11.6197

1.4000 3.5503 3.4000 5.2553 5.4000 8.1302 7.4000 9.6767 9.4000 11.4938

1.5000 3.7253 3.5000 5.3606 5.5000 8.2333 7.5000 9.9481 9.5000 11.6236

1.6000 3.8713 3.6000 5.4129 5.6000 7.7892 7.6000 10.0526 9.6000 11.6833

1.7000 4.0133 3.7000 5.5222 5.7000 7.9558 7.7000 9.9804 9.7000 11.6929

1.8000 4.1584 3.8000 5.6811 5.8000 8.3626 7.8000 9.8558 9.8000 11.6684

1.9000 3.9563 3.9000 5.9759 5.9000 8.1980 7.9000 10.0483 9.9000 11.8299

2.0000 4.3883 4.0000 6.1664 6.0000 8.1528 8.0000 10.8018 10.0000 11.8221

http://t2.lanl.gov/data/

http://t2.lanl.gov/data/


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Interaction of Charged Particles with Matter