Interaction of Heavy Charged Particles with MatterBAEN-625Advances in Food Engineering

Heavy charged particles

Charged particles other than the electron and positron

Energy-loss mechanisms

A heavy particle traversing matter loses energy primarily thru the ionization and excitation of atoms◦ Except in low velocities, it loses a negligible amount of

energy in nuclear collisionsThe moving particle exerts electromagnetic forces on atomic electrons and impart energy on themThe transferred energy may be sufficient to knock an electron out of an atom and thus ionize itOr it may leave the atom in an excited state

Heavy charged particleCan transfer only small fraction of its energy in a single electronic collisionIts deflection in the collision is negligibleThus it travels an almost straight path thru matter,It loses energy continuously in small amounts thru collisions with atomic electrons

Maximum Energy Transfer in a Single Collision

Assume ◦ the particle is moving rapidly compared to the

electron◦ Energy transferred is large compared with the

BE (binding energy) of the electron in the atom◦ The electron is free and at rest

Maximum Energy Transfer in a Single Collision

Maximum Energy Transfer in a Single Collision

Since energy and momentum are conserved

Solving for V1:

11

21

21

2

21

21

21

mvMVMV

mvMVMV

+=

+=

22

12

max

1

)(4

21

21

)(

mMmMEMVMVQ

mMVmMV

+=−=

+−

=E=MV2/2Initial KE

Incident Particle is an Electron

Its mass is the same as that of the struck particle, M = m

Entire energy can be transferred in a single, billiard-ball-type collision

EM

EMQ

MMMME

mMmMEQ

==

+=

+=

2

2

max

22max

44

)(4

)(4

Energy Transfer in a Single Collision if Incident Particle is an Electron

Maximum Energy Transfer in a Single Collision- Relativistic Expression

An electron is nonrelativistic as long as T is small compared with the rest energy, mc2 = 0.511MeV

cV

MmMmmVQ

/1/1

//212

2

22

22

max

=

−=

++=

ββγ

γγ

Qmax in Proton Collision with Electron

Proton Kinetic Energy, E

[MeV]

Qmax [MeV]

Maximum % Energy Transfer

100Qmax/E0.1 0.00022 0.22

1 0.0022 0.22

100 0.0219 0.22

100 0.229 0.23

1000 3.33 0.33

10000 136 1.4

100000 1060 10.6

1000000 53800 53.8

10000000 921000 92.1

Elastic or InelasticEquations shown before for Qmax are kinematic in natureThey follow from simultaneous conservation of momentum and KEThe assumption made to calculate energy loss was that the struck electron was not boundThus the collision being elasticCharged-particle energy losses to atomic electrons are, in fact, inelastic

Elastic Collision

Elastic collisionBoth momentum and kinetic energy are conserved This implies that there is no dissipative force acting during the collision and that all of the kinetic energy of the objects before the collision is still in the form of kinetic energy afterwardFor macroscopic objects which come into contact in a collision, there is always some dissipation and they are never perfectly elastic In atomic or nuclear scattering, the collisions are typically elastic because the repulsive Coulomb force keeps the particles out of contact with each other.

Examples of Elastic Collision

For a head-on collision with a stationary object of equal mass, the projectile will come to rest and the target will move off with equal velocity, like a head-on shot with the cue ball on a pool table. This may be generalized to say that for a head-on elastic collision of equal masses, the velocities will always exchange.

Examples of Elastic Collision

In a head-on elastic collision where the projectile is much more massive than the target, the velocity of the target particle after the collision will be about twice that of the projectile and the projectile velocity will be essentially unchanged.

Examples of Elastic Collision

In a head-on elastic collision between a small projectile and a much more massive target, the projectile will bounce back with essentially the same speed and the massive target will be given a very small velocity.

Inelastic Collision

Perfectly elastic collisions are those in which no kinetic energy is lost in the collision.Macroscopic collisions are generally inelastic and do not conserve kinetic energy, though of course the total energy is conserved. The extreme inelastic collision is one in which the colliding objects stick together after the collision

Inelastic Collision

Single-Collision Energy-Loss Spectra

Details about charged-particle penetration are embodied in the spectra of single-collision energy losses to atomic electronsThe collisions by which charged particles transfer energy to matter are inelasticKE is lost in overcoming the BE of the struck electrons

Single-Collision Energy-Loss Spectra

The ordinate gives the probability density W(Q)W(Q)dQ is the probability that a given collision will result in an energy loss between Q and Q + dQ

0 50 1000.00

0.06

0.04

50-eV electrons

5-MeV protons150-eV electrons

Energy Loss Q (eV)W

(Q) (

eV-1

)

In liquid water

Single-Collision Energy-Loss Spectra

For fast particles (speed > orbital speed)◦ Similarities in the region from 10-

70eVFor slow charged particles◦ The energy-loss spectra differ

from one another◦ The time of interaction is longer

than for fast particles◦ The BE is more important◦ Energy losses are closer to Qmax

◦ Slow particle excites atoms instead of ionizing them

A minimum energy Qmin >0 is required for excitation or ionization of an atom

0 50 1000.00

0.06

0.04

50-eV electrons

1-MeV protons

150-eV electrons

Energy Loss Q (eV)

W(Q

) (eV

-1)

In liquid water

Stopping PowerThe average linear rate of energy loss of a heavy particle in a medium [MeV/cm]Also referred as linear energy transfer (LET) of the particle

Stopping Powers

Can be calculated from energy-loss spectraFor a given type of charged particle at a given energy, the SP is given by◦ The probability μ per unit distance of travel that an electronic

collision occurs

◦ The average energy loss per collision, Qmax

∫

∫==−

=

max

min

max

min

)(

)(

Q

Qavg

Q

Qavg

dQQQWQdxdE

dQQQWQ

μμ

[MeV/cm][1/cm] [MeV]

Stopping Power-Semi Classical Calculation

YV

X

r

Fy

Fx m -e

ze

2

20

rzekF =

Coulomb force

Representation of the suddenCollision of a heavy chargedParticle with an electron,Located at the origin XY

θ

b

Stopping Power-Semi Classical Calculation

The total momentum imparted to the electron is the collision is:

22

420

2

20

02/12222

0 3/22220 32

22

0

22

2

2)(

2

)(22cos

b/rcosaxis)-Y cross particleheavy thetime( 0

coscos

bmVezk

mpQ

Vbzekp

VbtVbbtb

tVbdtbdt

rbdt

r

t

dtr

zekdtFdtFp y

==

=

=⎥⎦

⎤⎢⎣

⎡+

=

+==

==

===

∞

∞∞∞

∞−

∞

∞−

∞

∞−

∞

∞−

∫∫∫

∫∫∫

θθ

θθ

Stopping Power-Semi Classical Calculation

In traversing a distance dx in a medium having a uniform density of nelectrons per unit volume

The heavy particle encounters 2πnb db dx electrons at impact parameters between b and b + db

The energy lost to these electrons per unit distance traveled is 2πnQb dbThe total linear rate energy loss is:

min

max2

4220

2

4220

ln4

4d2 max

min

max

min

bb

mVnezk

dxdE

bdb

mVnezkbQbn

dxdE b

b

Q

Q

π

ππ

=−

===− ∫∫

Relativistic Stopping Power (Bethe’s Equation)

The linear rate of energy loss to atomic electrons along the path of a heavy charged particle in a medium is the basic physical quantity that determines the dose that the particle delivers in the medium

⎥⎦

⎤⎢⎣

⎡−

−=− 2

2

22

22

4220

)1(2ln4 β

ββ

βπ

Imc

mcnezk

dxdE

medium theofenergy excitationmean c torelative particle theof speed V/c light; of speed c mass;rest electron

medium in the eunit volumper electrons ofnumber charge;electron of magnitude particle;heavy theofnumber atomic

1099.8 2290

=====

===

×= −

Imn

ezCNmk

β

Stopping PowerDepends only on the charge ze and velocity β of the heavy particleThe relevant properties of the medium are its mean excitation energy I and the electronic density nm is the mass of the target atomic electronsUnits: MeV/cm, mass stopping power-[MeV cm2/g]

Stopping Power, general

For any heavy charged particle in any medium

22

26

2

231

1101.02ln)F(

MeV/cm ],ln)([1009.5

ββ

ββ

ββ

−−×

=

−×

=−−

eVIFnzdxdE

Mass Stopping PowerUseful quantity because it express the rate of energy loss of the charged particle per g/cm2 of the medium traversed

In gas –dE/dx depends on pressure, but –dE/ρdx does notMSP does not differ greatly for materials with similar atomic composition (primarily light elements)

For 10 MeV protons the MSP of H2O is 45.9 MeV cm2/g and for C14O10 44.2 cm2/g, however for Pb(Z=82) the MSP is 17.5 cm2/g

Heavy elements are less efficient on a g/cm2 basis for slowing down heavy charged particles (many of their electrons are too tightly bound in the inner shells to participate effectively in the absorption of energy)

Mean Excitation EnergiesCan be calculated from SP equationIt is the material parameter describing the ability of the target system to absorb energyEmpirical expressions:

⎪⎩

⎪⎨

⎧

>+≤≤+

=≅

13 71.88.52132 ,711211

1 Z ,0.19

eV, ZZZZ eV..

eVI

Mean Excitation EnergiesWhen a material is a compound or mixture, the SP can be calculated by simply adding the separate contributions from the individual componentsIf there is Ni atoms/cm3 of an element with atomic number Zi and mean excitation Ii:

∑=i

iii IZNIn lnln

Total # of electrons/cm3 in the material (n=ΣNiZi)

Example

Calculate the mean excitation energy of H2O

eVI

InZNI

eVIZOeVIZH

iii

O

H

6.74

312.4105ln10

810.19ln10

12lnln

10587.112.11),8(0.19),1(

=

=×

+×

==

=×+====

Table for Computation of SP’s

Use of table to facilitate the computation of SP for heavy charged particles

22

26

2

231

1101.02ln)F(

MeV/cm ],ln)([1009.5

ββ

ββ

ββ

−−×

=

−×

=−−

eVIFnzdxdE

Table for Computation of SP’sProton KE

[MeV]β2 F(β)

0.01 0.000021 2.1790.06 0.000128 4.8730.1 0.000213 5.1610.4 0.000852 6.7711 0.002129 7.6856 0.01267 9.75310 0.02099 9.97260 0.1166 11.96100 0.1834 12.16

SP of Water for Protons

Protons, Z = 1MW of water = 18.0g/molNumber of electrons/molecule, n = 10 ele/mol1 m3 of water = 106 gDensity of electrons is

3296

23 1034.3100.18

101002.6 −×=×××= mggn

SP of Water for Protons

Ln Iev = 4.3212SP of water for a proton of speed β is:

[ ]

[ ] 1

2

12

27031.469.700213.0170.0

69.7)(;00213.0 - Table - MeV 1at

31.4)(170.0

−

−

=−=−

==

−=−

MeVcmdxdE

F

MeVcmFdxdE

ββ

ββ

SP of Water for Any Particle

Previous table can be used for any other particle – get F(β) and β2 by using the following relationship:

proton

partic

proton

partic

MM

TT

=

Example

What is the SP for an 10-MeV alpha particle in water

dE/dx- calculate5.2 Table from and )F( find

5.24/10

414

2ββ

MeVMeVT

TT

proton

proton

alpha

==

==

Range

Of a charged particle is the distance it travels before coming to restThe reciprocal of the stopping power gives the distance traveled per unit energy lossSo, R(T) of a particle of kinetic energy T is:

∫

∫

=

⎟⎠⎞

⎜⎝⎛−=

−

T

T

GdE

zTR

dEdxdETR

02

0

1

)(1)(

)(

β

RangeFor a heavy particle:

)()(2

ββ pRzMR =

The proton rangeParticle’s velocity

Example

Use Table 5.3 to find the range of an 80-MeV 2He3+ ion in soft tissue

)(43)(

3,4

)()(

2

2

ββ

ββ

p

p

RR

Mz

RzMR

=

==

=

Example

esoft tissu

2

2

/529.0

529.0705.043)(

705.05.3 Tablein ion interpolat

7.263

80EnergyProton

)(43)(

ρ

β

ββ

cm

gcmR

cmR

MeV

RR

p

p

−

−

==

=

==

=

Ranges in cm of protons, alpha, and electrons in air at STP

For alpha particle at 15oC and 1 atm:

R in cm and E in MeV

84 ,62.224.14 ,56.0

<<−=<=

EEREER

Energy (MeV)10-2 10-1 100 101 102 103

10-1

100

101

102

103

104

105

Alphaparticles

protons

electrons

Ran

ge in

air

(cm

)

Slowing-Down Time

We can use the SP formula to calculate the rate at which a heavy charged particle slows down

The time rate of energy loss, -dE/dt can be expressed in terms of the SP by:

1111019.4

in water5.0h proton witfor

−×=−

=

⎟⎠⎞

⎜⎝⎛−=⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−=−

MeVsdtdE

MeVTdxdEV

dxdt

dxdE

dtdE

Slowing-Down Time

A rough estimation can be made of the time it takes a heavy charged particle to stop in matter, if one assumes that the slow-down rate is constant

sMeVsMeV

MeVTdtdEV

TdtdE

T

12

111

102.1)1019.4/()5.0(

in water5.0h proton witfor )/(/

−

−

×≈

×≈

=−

=−

≈

τ

τ

τ

Limitations of Beth’s EquationIt is valid at high energies as long as γm/M<<1 holds (e.g. up to ~ 106MeV for protonsAt higher energies it needs to consider◦ Forces on the atomic electrons due to the particle’s spin and

magnetic moment

It is based on the assumption that the particle moves much faster than atomic electrons ◦ At low energies it fails because the term 2mc2β2/I becomes

negative (given a negative value for stopping power)

many more (see text)