General Information

Muon Lifetime Update The next steps

Organize your results Analyze, prepare plots, fit lifetime distribution Prepare report using the Latex templates from the web site

Reports due May 14

Today’s Agenda Interaction of Particles with Matter (Summary) Cherenkov and Transition Radiation Interaction of Photons with Matter

The concept of cross sectionsCross sections or differential cross sections ddare used to express the probability of interactions between elementary particles.

Example: 2 colliding particle beams beam spot area A

= N1/t = N2/t

What is the interaction rate Rint. ?

Rint · t) = · L

Luminosity L [cm-2 s-1]

has dimension area !Practical unit: 1 barn (b) = 10-24 cm2

incident beam

solid angle element d

scattered beam

target

.nA = area density of scattering centers in target

Nscat() Ninc· nA · d= dd() · Ninc·nA· d

dE/dx ReviewHow do charged particles loose energy in matter ? Discrete collisions with the atomic electrons of the absorber material.

Collisions with nuclei not important (me

Average Energy Loss

22ln14 2max2

222

21

2222

T

Icm

AZzcmrN

dxdE e

eeA

dE/dx in [MeV g-1 cm2]

Bethe-Bloch formula only valid for “heavy” particles (mm).

dE/dx depends only on independent of m !

First approximation: medium simply characterized by Z/A (~ electron density)

Z/A~0.5

Z/A = 1

2

1

dxdE

22ln dxdE

“relativistic rise”

“kinematical term” 3-4minimum ionizing particles, MIPs

“Fermi plateau”

Minimum Ionizing Particles (MIPs)

-1 -1 2

min min

-3

Absorber MeVcm MeVg cm( )

Water 2.03 2.03Xenon (gaseous) 7.3 10

dE dEdx d x

1.24Iron 11.7 1.48Lead 12.8 1.13Hy -4drogen (gaseous) 3.7 10 4.12

Energy loss of minimum ionising particles

has broad minimum around = 0.96 or ~ 4Relativistic particles with an energy loss corresponding to this minimum are calledMinimum Ionizing Particles or MIPs.For a light absorber with Z/A ~ 0.5

-dE/dxmin ~ 2 MeV/(g/cm2)

Example

Bremsstrahlung Electrons and positrons lose energy via ionization just like other charged particles.

Small changes to calculation (identical particles, m(target) = m(projectile))

BUT dominant energy loss mechanism for high energy electrons is electromagnetic radiation

Circular acceleration: Synchrotron Radiation Motion through matter: Bremsstrahlung

ln522

122

1

22

41

2

kMv

kr

Mvcm

ZZc

edkd ee

Semi-classical calculation yields:

Cross section depends on•Incident particle’s mass (1/M2)•Medium (Z2)

Proportional to Z2/A of the Material.

Proportional to Z14 of the incoming particle.

Proportional to of the material.

Proportional 1/M2 of the incoming particle.

Proportional to the Energy of the Incoming particle

E(x)=Eo e(-x/X0) – ‘Radiation Length’

X0 M2A/ ( Z14 Z2)

X0: Distance where the Energy E0 of the incoming particle decreases E = E0e-1 = 0.37E0 .

Bremsstrahlung, QM

Radiation Length (Lr)The radiation length is a very important quantity describing energy loss of electronstraveling through material. We will also see Lr when we discuss the mean free path forpair production (i.e. e+e-) and multiple scattering.

There are several expressions for Lr in the literature, differing in their complexity.The simplest expression is:

)/)(183ln(4 23/121 AZZNrL aer

Leo and the PDG have more complicated expressions:

)/)1()](()183[ln(4 3/121 AZZZfZNrL aer Leo, P41

)]))(([4 21221

radradaer ZLZfLZNrL PDG

Lrad1 is approximately the “simplest expression” and Lrad2 uses 1194Z-2/3 instead of 183Z-1/3, f(z) is an infinite sum.Both Leo and PDG give an expression that fits the data to a few %:

)()/287ln()1(

4.716 2

cmgZZZ

ALr

The PDG lists the radiation length of lots of materials including: Air: 30420cm, 36.66g/cm2 teflon: 15.8cm, 34.8g/cm2H2O: 36.1cm, 36.1g/cm2 CsI: 1.85cm, 8.39g/cm2Pb: 0.56cm, 6.37g/cm2 Be: 35.3cm, 65.2g/cm2

Leo also has a table ofradiation lengths on P42but the PDG list is more up to date and larger.

Critical Energy: If dE/dx (Ionization) = dE/dx (Bremsstrahlung)

Muon in Copper: p 400 GeVElectron in Copper: p 25 MeV

W. Riegler/CERN 10

For the muon, the second lightest particle after the electron, the critical energy is at 400GeV.

The EM Bremsstrahlung is therefore only relevant for electrons at energies of past and present detectors.

Critical Energy

Z2 electrons, q=‐e0

W. Riegler, Particle Detectors

Interaction with the atomic electrons. The incoming particle looses energy and the atoms are excited or ionized.

Interaction with the atomic nucleus. The particle is deflected (scattered) resulting in multiple scattering of the particle in the material. During these scattering events a Bremsstrahlung photons can be emitted.

In case the particle’s velocity is larger than the velocity of light in the medium, the resulting EM shockwave manifests itself as Cherenkov Radiation. When the particle crosses the boundary between two media, there is a probability of the order of 1% to produce an X ray photon, called Transition radiation.

Electromagnetic Interaction of Particles with Matter

M, q=Z1 e0

Cherenkov Radiation

A charged particle travels through a medium at a speed larger than the local speed of light

Roger Forty Particle ID (Lecture I) 13

Cherenkov light

• Named after the Russian scientist P. Cherenkov who was the first to study the effect in depth (he won the Nobel Prize for it in 1958)

• From Relativity, nothing can go faster than the speed of light c (in vacuum)

• However, due to the refractive index n of a material, a particle can go faster than the local speed of light in the medium cp = c/n

• This is analogous to the bow wave of a boat travelling over wateror the sonic boom of an aeroplane travelling faster than the speed of sound

Roger Forty Particle ID (Lecture I) 14

Propagating waves

A stationary boat bobbing up and down on a lake, producing waves

Roger Forty Particle ID (Lecture I) 15

Propagating waves

Now the boat starts to move, but slower than the waves

• No coherent wavefront is formed

Roger Forty Particle ID (Lecture I) 16

Propagating waves

Next the boat moves faster than the waves• A coherent wavefront is formed

Roger Forty Particle ID (Lecture I) 17

Propagating waves

Finally the boat moves even faster• The angle of the coherent wavefront changes

cos = vwavevboat

Roger Forty Particle ID (Lecture I) 18

Speed calculation

• Using this construction, we can determine (roughly) the boat speed:

70º, vwave = 2 knots on water→ vboat = vwave/cos 6 knots

• Cherenkov light is produced when charged particle (vboat= c) goes faster than the speed of light (vwave= c/n)

→ cos C = 1 / n

• Produced in three dimensions, so the wavefront forms a cone of light around the particle direction

• Measuring the opening angle of cone → particle velocity can be determined

º

For Ne gas (n = 1.000067)

Nov 2004 19

Wave front comes out at certain angle

Cherenkov Radiation (2)

1cos c n

Threshold: > 1/n

Threshold Momentum for Cherenkov RadiationExample: Threshold momentum for Cherenkov light:

nt1

1

111

1222

nn

n

tt

t

)1)(1(1

112

nnn

tt

Example: Thresholds for different particles in He

)2(1

tt

The momentum (pt) at which we get Cherenkov radiation is:

)2(

mmp ttt

For a gas +2 so the threshold momentum can be approximated by:

2mmp ttt

For helium =3.3x10-5 so we find the following thresholds:electrons 63 MeV/c kaons 61 GeV/cpions 17 GeV/c protons 115GeV/c

Medium =n-1 thelium 3.3x10-5 123CO2 4.3x10-4 34H2O 0.33 1.52glass 0.46-0.75 1.37-1.22

For gases it is useful to set = n-1

Nov 2004 21

Cherenkov Radiation (3)

How many Cherenkov photons are detected?

For He we find: 2-3 photons/meter (not a lot!)For CO2 we find: ~33 photons/meterFor H2O we find: ~34000 photons/meter

We can calculate the number of photons/dx by integrating over the wavelengths thatcan be detected by our phototube (1, 2):

]11[sin2sin221

22

22

1

ddxdN

For a highly relativistic particle going through a gas the above reduces to:

photons/cm)1(780 ndxdN

GAS

Photons are preferentially emitted at small ’s (blue)

Nov 2004 22

Different Cherenkov Detectors

Threshold Detectors Yes/No on whether the speed is β>1/n

Differential Detectors βmax > β > βmin

Ring-Imaging Detectors Measure β

Nov 2004 23

Threshold Counter

Particle travel through radiator Cherenkov radiation

Types of Cerenkov Counters

Differential Cerenkov Counter:Makes use of the angle of Cerenkov radiation and only samples light at certain angles.For fixed momentum cos is a function of mass:

Not all light will make it to phototube

nppm

Epnn

22

)/(11cos

Differential cerenkov counters typically on work over a fixed momentum range(good for beam monitors, e.g. measure or K content of beam).

Problems with differential Cerenkov counters:Optics are usually complicated.Have problems in magnetic fields since phototubes must be shielded from B-fields

above a few tenths of a gauss.

Nov 2004 25

Ring Imaging Detectors (1)

Ring Imaging Cerenkov Counters (RICH)RICH counters use the cone of the Cerenkov light.The ½ angle () of the cone is given by:

nppm

n

2211 cos1cos

The radius of the cone is: r=Ltan, with L the distance to the where the ring is imaged.L

r

For a particle with p=1GeV/c, L=1 m, and LiF as the medium (n=1.392) we find:deg r(m)

43.5 0.95K 36.7 0.75P 9.95 0.18

Thus by measuring p and r we can identify what type of particle we have.Problems with RICH:

optics very complicated (projections are not usually circles)readout system very complicated (e.g. wire chamber readout, 105-106 channels)elaborate gas systemphoton yield usually small (10-20), only a few points on “circle”

Great /K/p separation!

Super Kamiokande

SuperK

481 MeV muon neutrino produces 394 MeV muon which later decays at rest into 52 MeV electron. The ring fit to the muon is outlined. Electron ring is seen in yellow-green in lower right corner. This is perspective projection with 110 degrees opening angle, looking from a corner of the Super-Kdetector (not from the event vertex). Color corresponds to time PMT was hit by Cerenkov photon from the ring. Color scale is time from 830 to 1816 ns with 15.9 ns step. In the charge weighted time histogram to the right two peaks are clearly seen, one from the muon, and second one from the delayed electron from the muon decay. Size of PMT corresponds to amount of light seen by the PMT. From: http://www.ps.uci.edu/~tomba/sk/tscan/pictures.html

SuperK is a water RICH. It uses phototubes to measure the Cerenkov ring.Phototubes give time and pulse height information

From SuperK site

SuperK has: 50 ktons of H2OInner PMTS: 1748 (top and bottom) and 7650 (barrel)outer PMTs: 302 (top), 308 (bottom) and 1275(barrel)

For water n=1.33For =1 particle cos=1/1.33, =41o

880.P20 Winter 2006 Richard Kass 29

The BaBar DIRCHere the challenge is to separate ’s and K’s in the range: 1.7

880.P20 Winter 2006 Richard Kass 30

The BaBar DIRC

1.5 T Solenoid Electromagnetic Calorimeter

(EMC)Detector of Internally

Recflected Cherenkov

Light (DIRC)

Instrumented Flux Return

(IFR) Silicon Vertex Tracker (SVT)

Drift Chamber (DCH)

phototube array

880.P20 Winter 2006 Richard Kass 31

Performance of the BaBar DIRCTiming information very useful to eliminate photons not associated with a track

±300 nsec window500-1300 background hits

±8 nsec window1-2 background hits

Note: the pattern of phototubes withsignals is very complicated. Thedetection surface is toroidal and thereforethe cerenkov rings are disjoint and distorted.

Use a maximum likelihood analysis to separate /K/p: L=L(c, t, n)DIRC works very well!

Z2 electrons, q=‐e0

4/18/201232

Transition Radiation

M, q=Z1 e0

When the particle crosses the boundary between two media, there is a probability of the order of 1% to produced and X ray photon, called Transition radiation.

Transition RadiationProduced by relativistic charged particles when they cross the interface of two media of different dielectric constants (Note that n ~ sqrt())

Qualitative Explanation:Since the electric field of the particle is different in the two media, the particle has to “shake off” the difference when it crosses the boundary. The total energy loss depends on the Lorentz factor= E/mc2

Mostly forward directed

Intensity roughly proportional to the energy E

Typically X-ray photons with energies between 5 –15 keV

The number of photons produced is very small. About 0.8 photons per transition for a particle with = 2000 (highly relativistic)

Stack foils to increase number of transitions

Interactions of Photons with Matter

There are three main contributions to photon interactions:Photoelectric effect (E < hundreds of keV)Compton scattering (Medium energies ~ MeV)Pair production (dominates at energies > few MeV)

A beam of ’s with initial intensity N0 passing through a medium is attenuated in number (but not energy) according to:

dN=-Ndx or N(x)=N0e-x

With = linear attenuation coefficient which depends on the total interaction cross section (total= coh+ incoh + +).

Intensity:

Interaction of photons

...0

pairComptonphoto

xeII

: mass attenuation coefficient

gcmA

Ni

Ai /

2

1 M

eV

photo effect

Rayleigh scattering(no energy loss !)

Compton scattering

pair production

Interaction of Photons

Thomson and Rayleigh Scattering No energy transfer (just change in photon direction) Low energies Rayleigh scattering off the atom as a whole (coherent effect)

Photo Effect Low energy (~ binding energy of electrons in atoms) Higher cross section for high Z material (~ Z4-5)

Compton Scattering Medium energies Klein Nishina formula

Compton edge (maximum recoil energy)

Pair Production E > 1.022 MeV

21

2max hT

2

222

22

cos11cos1cos1

cos1112

cmhwithr

dd

ee

Photon Conversions

Otherwise known as pair production.

Threshold: 2mec2 (nucleus) 4mec2 (atomic electron)

Total cross section increases rapidly with photon energy, approximately proportional to Z2.

Comparing pair production with bremsstrahlung:

Or for the mean free path:

pair 7 9 brem

Electromagnetic Showers

• a beam of electrons impinging on solid matter will have a linear absorption coefficient of 1/X0

• this process repeats, giving rise to an e.m. shower:

• the process continues until the resulting photons and electrons fall below threshold

• so how do we get some sort of signal out?• ultimately we need ionizationWill discuss more when we talk about calorimetry…

Basic EM Interactions

e+ / e-

IonizationdE/dx ~ 1/2, z2

BremsstrahlungdE/dx ~ 1/m2, z4

Photoelectric effect

Compton effect

Pair production

E

E

dE/d

x

E

dE/d

x

E

E