Lectures 9-10

24-25/11/05, Lect. 9-10 Nuclear Physics Lectures, Dr. Armin Reichold 1

Lectures 9-10Particle interactions with

matter

24-25/11/05, Lect. 9-10 Nuclear Physics Lectures, Dr. Armin Reichold

2

9.0 Overview 9.1 Introduction 9.2 Charged particles in

matter Classification of

interactions Non-radiating interactions

(ionisation) Radiating interactions

Ionisation and the Bethe-Bloch formula (BBF)

End of lecture 9

Radiating interactions Cherenkov-radiation Bremsstrahlung Synchrotron-radiation

The em-shower

9.3 Photons in matter Overview

Photoelectric effect Raleigh scattering Compton scattering Pair production

Comparison of cross-sections

Z & A dependence of cross-sections

Similarities between pair production and Bremsstrahlung


3

12.1 Introduction(why do we need to know this)

Measure properties of nuclei through decay products Measure energy, momentum, mass & charge of particles

with M [0 () ; few 100 GeV (fission fragment)] Ekin [keV (Radioactivity) ; few GeV (accelerator experiments)] Q/e [0 (,n); O(100) (fission fragments)]

Need to translate microscopic particle properties into quantitatively measurable macroscopic signals

Do this by interactions between particles and matter Which interactions would be useful?

Weak? Too weak at low (nuclear) interaction energies Strong? Some times useful but often noisy (strong fluctuations, few

interactions per distance) EM? Underlies most nuclear and particle physics detectors (L9&10)

Energies released ≤ Ekin(particle) often too small for direct detection need amplification of signals (see detector section L11)


4

12.1 Introduction Particle Ranges

a) If smooth energy loss via many steps (i.e. ionisation from light ions)

sharply defined range, useful for rough energy measurement

a)

b)

c)c) Sometimes several types of processes happen (i.e. high energy electrons)

mixed curves, extrapolated maximum range

b) If a few or a single event can stop the particle (i.e. photo-effect)

exponential decay of particle beam intensity, decay constant can have useful energy

dependence No range but mean free path defined


5

12.1 Introduction Particles we are interested in

photons exponential attenuation at low E, often get absorbed in single events detect secondary electrons and ions liberated in absorption process.

charged particles sharper range (continuously loose energy via ionisation) leave tracks of ionisation in matter measure momentum in B sometimes radiate photons can be used to identify particle type

neutrons electrically neutral no first-order em-interaction devils to detect react only via strong force (at nuclear energies!) long exponential range (lots of nuclear scattering events followed by

absorption or decay) need specific nuclear reactions to convert them into photons and/or

charged particles when captured by a target nucleus if stopped, measure decay products, e- + p +


6

12.2 Charged particles in matter(classification of interactions)

If particle or medium emit photons, coherent with incoming particle radiation process

Bremsstrahlung, Synchrotron-radiation: emitted by particle

Cherenkov-radiation: emittted by medium If no coherent radiation non-radiating process

Ionisation, scattering of nuclei or atoms Note: Scintillation is a secondary process in

which the light is emitted after ionisation or atomic excitation. It is NOT a radiation process


7

12.2 Charged particles in matter(non radiating interactions, what to collide with)

What could a charged particle collide with Atomic electrons (“free”)

large energy loss E≈q2/2me (small me, q=momentum transfer) small scattering angle

Nuclei small energy loss (E=q2/2mnucleus) large scattering angle

Unresolved atoms (predominant at low energies) medium energy loss E<q2/2me

eff because: me

eff(bound)>me(free) medium scattering angle atoms get excited and will later emit photons (scintillation)


8

12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula)

Deal with collisions with electrons first since these give biggest energy loss.

Task: compute rate of energy loss per path length, dE/dx due to scattering of a charged particle from electrons in matter.

Remember a similar problem? Scatter alpha particles of nuclei =

Rutherford scattering


9

12.2 Charged particles in matter(Comparison between Rutherford Scattering and EM-scattering of free

electrons) Rutherford Scattering

any charged particle X (original used ’s) scatters of nucleus

Charge(X)=Ze Charge(nucleus)=Z’e Mnucl >> MX no nuclear-

recoil first order perturbation theory

(Z*Z’*em<<1) point point scattering

no form-factors

Bethe-Bloch situation any charged particle X

scatters of electron (in matter)

Charge(X)=Ze Charge(electron)=1e MX >> Me no X-recoil (not

true for X=e-) first order perturbation theory

(Z*1*em<<1) point point scattering

no form-factors

com

mon

aliti

esdi

ffere

nces spin-0 scatters of spin-0

non-relativistic nucleus assumed unbound

spin-0 scatters of spin-½ could be relativistic electron is often bound


10

12.2 Charged particles in matter(Comparison between Rutherford Scattering and EM-scattering of free

electrons)

Will initially ignore the spin and relativistic effects when deriving first parts of Bethe Bloch formula

Start with Rutherford like scattering using electron as projectile

Later introduce more realistic scattering crossection (Mott) to get full Bethe Bloch formula

Add effects for bound electrons at the end


11

12.2 Charged particles in matter(From Rutherford Scattering to the Bethe-Bloch Formula)

Differential Rutherford-scattering crossection for electrons as projectiles

If we want to turn this process around to describe energy loss of a particle X scattering of electrons in a solid we need to initially assume:

X scatters of free electrons i.e. Ekin,projectile >> Ebin,electron or Vprojectile>>Vbound-e (deal with bound electrons later)

MX>>me so that reduced Mreduced(X) ≈ Mrest(X) will need recoil corrections to apply results to dE/dx of electrons passing through matter

22 24

2 2 sin4 2Rutherford z cdd P V

h

P,V = momentum and relative velocity of electron wrt. nucleusZ = charge of nucleus = scattering angle of the electron wrt. incoming electron direction= stereo angle


12

2 22

2 2 21 2 sin2 sin

d d d d dq d d p dpd d d d d dq dq dq

2 2 2 2 cosq P P PP

12.2 Charged particles in matter(normal Rutherford Scattering: e- on nucleus, change of

variables)

Change variables from to q2 (q = momentum transfer to electron) to get to frame independent form

Pelectron,in

P’electron,out

q

2 2 2 2

in elastic scattering of heavy nucleus:

2 (1 cos ) 4 sin 2

P P p

q p p2

22 sindq pd

1if no dependence: 2 sin 2 sindd dd

2 1sin 1 cos2 2


13

2 2

22 2

2 2 4

since:

4

d ddq p d

z cddq V q

h

12.2 Charged particles in matter(normal Rutherford Scattering: e- on nucleus, change of

variables)

22 24

2 2 sin4 2z cd

d p Vh

22

2

22 2 2

2 4

since: sin 2 44

qp

z cd pd V q

h


14

12.2 Charged particles in matter(Rutherford Scattering, change of frame to nucleus on e)

Change frame to: electron stationary (in matter), nucleus moving

with V towards electron p in formula is still momentum of electron

moving with relative V p =meV q2 is frame independent

non-relativistic this is obvious (do it at home) Energy transfer to the electron is defined via: E==|q2|/2me d/dq2=1/2me

relativistic need to define q as 4-momentum transfer, but we assume non relativistic for Rutherford anyway.


15

Above is crossection for a non relativistic heavy particle of charge z to loose energy between and +d in collision with a spin-less electron it approaches with velocity V

We want as a useful quantity: kinetic energy lost by projectile = -dT per path length dx in material of atomic number density n with Z’ electrons per atom

12.2 Charged particles in matter(From inverse Rutherford scattering to the Bethe-Bloch Formula)

22 2

2 22 1

e

z cd d dd mV

h

max

minv

ddT nZdx dd

number of collissions withelectrons in length dx per unit crossection area

crossection weightedavg. energy lost per collision

22 2

2 2 2 441

2 e

z cd d d ddq dq d m d V q

h

|q2|=2me


16

12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, simple integral)

max

min

2 2 2 2 2 2max

2 2 2min

1 2 ( ) 2 ( ) lne e

dT Z c Z cnZ d nZdx mV mV

h h

Two of our assumptions justifying the use of Rutherford scattering were:

Electrons in matter have no spin Projectile travels at non relativistic speed

None of these are met in practise We have to do all of the last 5 slides again

starting from a relativistic crossection for spin ½ electrons.


17

12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, Mott)

Differential Mott-scattering crossection for relativistic spin ½ electrons scattering off a finite mass nucleus (finite mass e- could be target)

22 2 24 2

2 2 2sin 1 sin4 2 2Mott

z cd Vd P V c

h

If we perform the same transformations (q2) with this crossection and then perform the integral:

Rutherford term Mott term

max

min

Mott

v

ddT nZdx dd

we get …

18

12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, Mott integral)

Valid for all charged particles (not limited to heavy particles)

max can be computed via kinematics of “free” electron since Ebind << Ekin (see Williams problem 11.1 on p.246)

2 2 2

max 22

1 2

with: and as properties of incoming particle

e

e e

m cm mM M

E PM E

max 2

if incoming particle is not relaticistic ( 1) and

4( )

e

e

E T M T Mm MT

M m

==

2 2

max 2

2

Note: c=1 from here downwards!2( )

2ee

E MM m Em

2

max

if and and then (ultra relativistic incoming particle)

e eE M E M m E mE

? ? ?

A list of limits for max follows:

2 2 2 2max max min

2 2 2min

2 ( ) ln 1 2e e

dT Z c VnZdx mV c m c

hMott term


19

12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, min)

But what about min ? can not assume that e is free for small energy transfers ≠q2/2me because electron bound to atom can get excited atoms in final state (not just ions) our integral was wrong for the lower limit! (can’t get

from first to second line on slide 15 any more) For small need 2-D integral d dq depending on

detailed atomic structure We need to find some average description of the atomic

structure depending only on Z and A if we want to find a universal formula

This gives sizable fraction of integral but is very hard to do

The result is the Bethe-Bloch Formula

20

12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula = BBF)

Stopping power = mean energy lost by ionisation upon perpendicularly traversing a layer of unit mass per area.

Units: Mev g-1 cm2, Range: 4.1 in H to 1.1 in U I=mean excitation energy; depends on atom type, I≈11*Z [eV]

2 2 2max

2 2 22 2 22

2

in the infinitely heavy projectile ("no recoil", 2 ) approximation =2 and this becomes:

21 4 ln 2

e

e

eA

e

M mm c

m cN ZdT Zdx A m I

?

h

21with , , mass density, density correction

1-atomic number, Avogadro's number, mean excitation potential A

Vc

A N I

1 is called Stopping Power.dTdx

2 2 22 2 22max

2 221 4 1ln2 2

eA

e

m cN ZdT Zdx A m I

h


21

12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, Bethe-Bloch features)

=density correction: dielectric properties of medium shield growing range of Lorenz-compacted E-field that would reach more atoms laterally. Without this the stopping power would logarithmically diverge at large projectile velocities. Only relevant at very large

BBF as a Function of is nearly independent of M of projectile except for max and very weak log dependence in if you know p and measure get M (particle ID via dE/dx): See slide 23

Nearly independent of medium. Dominant dependence is Z’/A ≈½ for most elements.

Limitations: totally wrong for very low V (ln goes negative particle gains Energy =

stupid) correct but not useful for very large V (particle starts radiating, see next

chapter)

2 2 22 2 22max

2 221 4 1ln2 2

eA

e

m cN ZdT Zdx A m I

h

1ln ln and 28.816 (2 2

pp Z A

I

(off syllabus)


22

12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, variation with )

+ can capture e-

Ec = critical energydefined via:dE/dxion.=dE/dxBrem.

BetheBloch

Broad minimum @ ≈3.0(3.5) for Z=100(7) At minimum, stopping power is nearly

independent of particle type and material

Stopping Power at minimum varies from 1.1 to 1.8 MeV g-1 cm2)

Particle is called minimum ionising (MIP) when at minimum


23

in drift chambergas

12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, variation with particle

type)

P=mv=mc variation in dE/dx is

useful for particle ID variation is most

pronounced in low energy falling part of curve

if you measured P and dE/dx you can determine the particle mass and thus its “name”

e


24

12.2 Charged particles in matter(Radiating Interactions)

Emission of scintillation light is secondary process occurring later in time.

Has no phase coherence with the incident charge and is isotropic and thus SCINTILLATION NOT A RADIATING INTERACTION in this sense.

Primary radiation processes which are coherent and not isotropic are:

Cherenkov radiation is emitted by the medium due to the passing charged particle.

Bremsstrahlung and Synchrotron Radiation are emitted by charged particle itself as result of its environment.


25

12.2 Charged particles in matter(Cherenkov Radiation)

Source of E-field (Q) passing through medium at a v > vphase(light in medium) creates conical shock wave. Like sonic boom or bow wave of a planing speed boat.

Not possible in vacuum since v<c. Possible in a medium when v>c/n. The Cerencov threshold at = 1/n can be used to measure and thus do

particle ID if you can measure the momentum as well. Huygens secondary wavelet construction gives angle of shockwave

as cos = 1/n, This can be used to measure particle direction and .

In time that the particle goes from O to P, light goes from O to A.

Cherenkov radiation first used in discovery of antiproton (1954).

Now often used in large water-filled neutrino detectors and for other particle physics detectors (see Biller).

Total energy emitted as Cherenkov Radiation is ~0.1% of other dE/dx.

ct/n

ctO P

A

particle trajectory


26

12.2 Charged particles in matter(Cherenkov Radiation)

Picture of Cherenkov light emitted by beta decay electrons in a working water cooled nuclear reactor.

27

12.2 Charged particles in matter(Bremsstrahlung = BS = Brake-ing Radiation)

Due to acceleration of incident charged particle in nuclear Coulomb field

Radiative correction to Rutherford Scattering. Continuum part of x-ray emission spectra. Electrons “Brem” most of all particles because

radiation ~ (acceleration)2 ~ mass-2. Lorentz transformation of dipole radiation from

incident particle frame to laboratory frame gives “narrow” (not sharp) cone of blue-shifted radiation centred around cone angle of =1/.

Radiation spectrum falls like 1/E (E=photon Energy) because particles loose many low-E photons and few high-E photons. I.e. It is rare to hit nuclei with small impact parameter because most of matter is “empty”

Photon energy limits: low energy (large impact parameter) limited through shielding

of nuclear charge by atomic electrons. high energy limited by maximum incident particle energy.

Ze

e- e-

e-*


28

12.2 Charged particles in matter(Bremsstrahlung EM-showers, Radiation length)

dT/dx|Brem~T (see Williams p.247, similar to our deriv. of BBF and plot on slide 22) dominates over dT/dx|ionise ~ln(T) at high T.

Ecrit = Energy at which BR-losses exceed ionisation losses (see slide 22)

For electrons Bremsstrahlung dominates in nearly all materials above few 10 MeV. Ecrit(e-) ≈ 600 MeV/Z

If dT/dx|Brem~T T(x)=T0 exp(-x/X0) Radiation Length X0 of a medium is defined as:

distance over which electron energy reduced to 1/e via many small BS-losses

X0 ~Z 2 approximately as it is the charge that particles interact with Bremsstrahlung photon can produce e+e--pair (see later) and start

an em-shower (also called cascade, next slide) The development of em-showers, whether started by primary e or

is measured in X0.


29

12.2 Charged particles in matter(simple EM-shower model)

Simple shower model assumes:

e≈2 E0 >> Ecrit only single Brem- or pair

production per X0 The model predicts:

after 1 X0, ½ of E0 lost by primary via Bremsstrahlung

after next X0 both primary and photon loose ½ E again

until E of generation drops below Ecrit

At this stage remaining Energy lost via ionisation (for e+-) or compton scattering, photo-effect (for ) etc.

Abrupt end of shower happens at t=tmax = ln(E0/Ecrit)/ln2 Indeed observe logarithmic dependence of shower depth on E0


30

12.2 Charged particles in matter(Synchroton Radiation)

Appears mainly in circular accelerators (mainly to electrons) and limits max. energy achievable.

Similar to Bremsstrahlung Replace microscopic force from E-field in Bremsstrahlung

with macroscopic force from vxB to keep electron on circular orbit

Electrons radiate only to the outside of circle because they are accelerated inward

Angle of maximum intensity of synchrotron radiation with tangent of ring =1/

Synchrotron radiation = very bright source of broad range of photon energies up to few 10 keV used in many areas of science

Many astrophysical objects emit synchrotron radiation from relativistic electrons in strong magnetic fields


31

13.1 Photons in matter(Overview-I)

Rayleigh scattering Coherent, elastic scattering on the entire atom (the blue sky) + atom + atom dominant at >size of atoms

Compton scattering Incoherent scattering on electron from atom + e-

bound + e-free

possible at all E > min(Ebind) to properly call it Compton requires E>>Ebind(e-) to approximate

free e-

Photoelectric effect absorption of photon and ejection of single atomic electron + atom + e-

free + ion possible for E < max(Ebind) + E(Eatomic-recoil, line width) (just above

k-edge)


32

13.1 Photons in matter(Overview-II)

Pair production absorption of in atom and emission of e+e- pair Two varieties:

a) dominant: + nucleus e+ + e- + nucleusrecoil b) weak: + Z*atomic e- e+ + e- + Z *atomic e-

recoil Both variants need: E>2mec2 + Erecoil

bigger Mrecoil gives lower threshold because Erecoil = Precoil2/2Mrecoil

type a) has lower threshold then type b) because Mnucl>>Meeff

Nucleus/atom has to recoil to conserve momentum coupling to nucleus/atom needed strongly charge-dependent crossection (i.e. growing with Z)

type a) has aproximately Z times larger coupling dominant

33

13.1 Photons in matter (Crossections)

R Rayleigh PE Photoeffect C Compton PP Pair Production on nucleusPPE Pair Production on atomic electrons PN Giant Photo-Nuclear dipole resonance

Carbon

Lead

As Z increases PE extends to higher E due to stronger atomic e- binding PP & PPE extend to lower E due to stronger coupling of projectile to target Threshold for PPE decreases as nucleus contributes more to recoil via stronger atomic

electron-nucleus bond As A increases Erecoil (nucleus) decreases and threshold for PP gets closer to minimum

of 2*mec2


34

13.1 Photons in matter(Comparison of Bremsstrahlung and Pair Production)

Very similar Feynman Diagram Just two arms swapped

Typical Lenth =Radiation LengthX0

Typical Lenth =Pair Production Length L0

L0=9/7 X0

Ze

e- e-*

Bremsstrahlung

e-

Ze

e-* e-

Pair production

e-

X0 :distance high E e-travels before it reduces its energy by 1/e or

E(e-)=E0*exp(-x/X0)X0=attenuation length

L0 :distance high E

travels before prob. for non

interaction reduced to 1/e

P()=1/L0*exp(-x/L0)

L0=mean free path

Lectures 9-10

Documents

Transcript of Lectures 9-10