8

Spring 2012, P627, YK Monday January 30, 2012

Observable particle detection effects are :

• (most) due to long‐range e‐m forces i.e. via atomic collisions• or due to short‐range nuclear collisions• or through decay ( = weak interactions)

Let’s first consider (1) interactions of charged particles with matter via e‐m.

Literature:• B. Rossi  “High Energy Particles” , Prentice‐Hall, Inc., Englewood Cliffs, NJ, 1952• J.D. Jackson, Classical Electrodynamics, Chapter 13• Print:  http://pdg.lbl.gov/2011/reviews/rpp2011‐rev‐passage‐particles‐matter.pdf

2

Swift particle with charge and mass ( . . , , , ...) with

and collides with atomic electron of mass

and charge ( ). Electrons are in atoms with , and .

Energy losses of

e

ze M e g p d

Mc p Mc m m

e Z A

m a

g gb= = =-

E

particle with mass M are from the collisions with atomic electrons.

In many cases binding energy of electrons in atoms can be

neglected for energetic collisions "scattering on free electrons"

Energy losses of all particles with ( , , , , , ...) can be described

by general law (from Rutherford scattering) with as a parameter (+spin)eM m K p d

M

m p a

Elastic on via Coulomb scattering Rutherford scattering f-laM m

General Disposition 

Energy losses by or particles are treated differently die to ee e M m+ - =

Electromagnetic interactions of neutal particles are not zero,

but very small (e.g. neutrons).

Energy loss X-section for scattering particles m1 and m2

22 2

2 2

m vTe

¢=

energy transferred in Lab from particle 1 to particle 2= energy lost by particle 1 in Lab 

2

2 max2 22

1 22 max 1 Lab2

1 2

2 , 0,

4

( )

Q dd E

m v

m mE E

m m

es p e

e¥

é ù¢= Î ë û

¢ =+

12

1 2

2sin LL1 (17.5)

2

mv v

m m

c¥¢ =

+

22 2

2

2sin

2

mv

m

ce ¥=

2

2 3

cos 2

sin 2

Qd d

mv

cs p c

c¥

æ ö÷ç ÷ç= ÷ç ÷÷çè øRutherford scattering for  ( )

QU r

r=

2

1d

d

se e

µ

21 is for Coulomb (= atomic) field.Compare with for e.g. nA low-energy scattering

d d conste

s e =

Scattering of particle with mass m and charge z on electrons in matter  (electrons are in atoms with Z and A) 

[4] B. Rossi  “High Energy Particles” , Prentice‐Hall, Inc., Englewood Cliffs, NJ, 1952  !

Energetic knock‐out of electrons (‐rays)

Energy loss X-section for scattering particles m1 and m2

22 2

2

m vT e

¢= = energy transferred in Lab from particle 1 to particle 2

= energy lost by particle 1 in Lab 

2

2 max22

1 22 max 1 La2

2

b1 2

2 , 0,

4

( )

Q dd E

m

m mE E

m

v

m

es p e

e¥

é ù¢= Î ë û

¢ =+

12

1 2

2sin LL1 (17.5)

2

mv v

m m

c¥¢ =

+

22 2

2

2sin

2

mv

m

ce ¥=

2

2 3

cos 2

sin 2

Qd d

mv

cs p c

c¥

æ ö÷ç ÷ç= ÷ç ÷÷çè øRutherford scattering ( )

QU r

r=

2 2

1 1 or

d d

d dT T

s se e

µ µ

this is non relativistic treatment,

– kinetic energy transferred to electron

= energy loss by incident particle

[Remember 2 is invariant]

T

t mT

-

= -

2

- probabilty distribution function (pdf) for charge particle with velocity passing the layer of thickness

d N

dTdx dxb

2[ ] usually expressed in units ( )dx g cm dx dlr= ⋅

2 2

2

4where 0.307075 A e eN r m cK MeV

A A g cm

p= =

for 1A g mol=

relativistic formula

( ) includes QFT spin treatment

of colliding particles

F T

2

max2 1

2 22 (Bhabha+Massey+Corber)max

2

max

(Bhabha)( ) 1 for incident spin 0 particle

for spin incident1( ) 1

2

1 1( ) 1 1

3 3C

TF T

T

T TF T

T Mc

T T TF T

T T Mc

b

b

b

é ùê ú= -ê úë ûé ùæ öê ú÷ç= - + ÷çê ú÷ç ÷è øê ú+ë ûæ öæ ö÷ ÷ç ç÷ ÷= - + +ç ç÷ ÷ç ç÷ ÷ç çè øè ø +

E

E

2

2

2 2

(Massey+Corber+Oppenheimer+...)

11

2

for incident spin 1 particle

C

C

T

T

M cT

m

é ùæ öæ öê ú÷÷ çç ÷÷ +ççê ú÷÷ çç ÷ ÷çè ø è øê úë û

2

20

;

2.824

classical electron radius;

charge of particle

e

e

v

ce

r fmm c

z

b

p

¥=

= =

-

relativistic formula from your HW

2 2 2max

2

since 1 and when is not very large the 2

and are ~"independent" on mass (but function of = or = )

m M T mc

d NM pc M

dTdx

g b g

b gE E

ø 2

22222

2 2

2

2222 2

2 2

For incident (M ller) for

1 ( )1 1

2 ( ) 2 ( )For incident (Bhabha) for

1 ( )1 2 2( )

2 ( )

e mc

T Td N Z K ZK

dTdx A T T A T Te mc

T Td N K ZT T

dTdx A T T

-

+

é ù- +é ù ê úë ûê ú= - = ⋅ê ú- -ë û

é ù- +ê úë û é ù= ⋅ - + ⋅ê úë û-

E

E EEE

E E EE

E EE E E

E

additional term due to possible positron and electron recoils

maxExercise: What is if incident particle is ?T e-

2 2 2 2 2 2 2

max _2

2 11

1 11 2 ( )kin incm m

M M

mc mcT T

b g b g b gg g gg

æ ö÷ç= = = = - ÷ =ç ÷ç ÷è ø+ ++ +

EE

max

However, in scattering electrons are not distinguishable

2

e e

T

- -

= E

maxF-la for positrons is treated as 2 as well !!!T = E

2 2

max

22

2 2

If T T and T T all above f-las are transform

1 1to Rutherford's formula:

2

CM c

mdN K Z

zdTdx A Tb

=

= ⋅ ⋅

2

1note when T 0

due to long-range of e-m forceT

¥

( )max 2

; (average ionozation potential as minimum transfer energy)

TdE d NT T dT I

dx dTdxe

e eæ ö÷ç ÷- > = ç ÷ç ÷÷çè øò

Bethe‐Bloch formula:

called “electronic stopping power” for M>m particles

2 2

accuracy of B-B f-la is about 1%here 0.3

in the "useful" energy range K MeV dE MeV

A dxg cm g cm

é ùê ú = ê úë û

For practical purposes in a given material is function of only ;

for all heavy particles same ( ) can be used;

dEdx

dE pdx Mc

bg

bg bg =

dx

1x 2x

1E E= 2E E E= -D

2

1

E= x

x

dEdx

dxD ò

Stopping  power and Range

2 2

1 1

= E E

E E

dx dEx dE

dE dEdx

D =æ ö÷ç ÷ç ÷ç ÷è ø

ò ò

0

( )( ) E

dER range E

dEdx

=æ ö÷ç ÷ç ÷ç ÷è ø

ò

Stopping power

Ranges

(a) http://physics.nist.gov/PhysRefData/Star/Text/contents.html

Stopping‐Power and Range Tables for Electrons, Protons, and Helium Ions:

(b) http://www.srim.org/

Average ionization potential = mean excitation energy is an important

non-trivial parameter that is detemined experimentally

I

compare with known

I=13.6 eV for hydrogen