Olav Ullaland / PH department / CERN

!"#

$ %$ &

$ ' &( ) * #+,-.-/

0 &

!1 %2 3) * #4,5554

6! "7 "8 " !

++9.

$

: ,..; -

< &"" "#==">!>> ="= "#==>!>=

7 % %""">

? "!&>

DisclaimerThe data presented is believed to be correct, but is not guaranteed to be so.

! " "# "$% $% & ' ( #&!

( ) ) ) & *

+=

+ &,+ &, -.+ &, -/

( ) <≤= ββ

( )∞<≤−

= γβ

γ

=== βγβγ

0, *0, 10*2 30.405 10 16 *730.

8/ ,-/

10-→ 10.48 12 */630.06 ,

"9 10:3 0.0/ ,

9 ) "9 & 0.0: × 0:3 0.0/ , ≅ 0.7

$; 0.. $0..$-

C. Joram, SSL 2003

*

σ &&σ-&Ω<*

Φ11% 0- Φ21% /-

/

0

' +

Lσ=⋅

∝

σ &*010.4/: /

4/40L

+&

=&

=

( ) ΩΦΘΩ

=

Ω∝ΦΘ

σ

C. Joram, SSL 2003

? @":#

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<

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) )$$

Turn of a century. 1900

Turn of another century. 2000

&

/ 4$$

5678$

#$& $

#. / #

. &9':

2#> 1000 tracks

The experi-mental set-ups are not what they used to be!

4

;<%$

:&=

0 2&%3

&&) ) * = && *

http://cmsdoc.cern.ch/ftp/TDR/TRACKER/tracker_tdr.html

& '>4'?'$) :

@ & $ ( )

# & &

2 A

2# $ #

#& $ #

#$

,'+ # %$#

%$# & # #

#$#

) & +

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< >&?&&*&$) ) * $) ) && <>?&) ) $* &< <*=<&&& @& <& $) &*

"%,.;+6!%# *( **AB ' *! CA/9 ( % (

? & 6# *$&"( ; ( ; 9 ; ( % (

<) D) '% *CERN photo 1955

DANGERCOSMIC

Anode

CathodeQ

=E &&*

( E

( >?*

"&*=α ) 0&&& @ α

=) *

αααα 6 *

) ( )

−= η

B && B && &&&&&

) &) &* $&$* &+

( )

−−=

α

α

γγγγγ ) && ******

E0

E1> E0

E2< E0

E3> E0+

-

&) &) &) *

S.C. Brown, Introduction to Electrical Discharges in Gases, 1966

& *

FFA

hd

r

&)

=&&

α 4 &$4&) &&&) ) * $& &

α

∝

1 ) &*

→ &A1&

→ &) ) &&*

→ &&) &&

&&&&&) &) $&) $*C *( $1" → &∼0.G -

A) & &*

) &$) ) &*

Rutherford Scattering (non relativistic)

b

m1, Z1e, v0

m2, Z2e

∆Θ∆Θ∆Θ∆Θ

r2

r1

r12

?

:

πε+=′′=′′−′′

=Θ

:πε=

+

=

Angular deflection is then

where

and b is the impact parameter

Integrating over b

[ ] ( )

:

:

5

Θ=

Ω επ

N0 number of beam particles n target material in atoms/volumet target thickness

Θcm > Θminsince there is a screening of the electric field of the atom.

#

α≈Θ where a0 is the first Bohr radius

-4 #&

. 8#BC?

D)

D)5

D):

D)?

@ #$#@78

Θ$

Θ

#

#2

#

ΘΘΘ≈Θ

for a single scattering

[ ]

≈Θ∗=Θ

β

απρ

Alpha particles in silver

1

10

100

1000

10000

100000

1000000

0 30 60 90 120 150 180

Mean angle of scattering

Relative number of

scattered

: Θ∝

!$& $$ & # &

$#$>

EF $ $&&&)

&$ $

3 $E

D!$& ;6$&

$@

$>@$$#

=

=

0

#2

:

πε

Multiple Coulomb Scattering (after Rutherford)Energy Transfer = Classic RutherfordIf Energy > Ionization Energy → electron escape atom

< → No energy transfer

where the sum is taken over all electrons in the atom for which the maximum energy transfer is greater than the ionization energy.

Substituting in the maximum (non relativistic) energy transfer:

Excitation energies (divided by Z) as adopted by the ICRU [Stopping Powers for Electrons and Positrons," ICRU Report No. 37

(1984)].Those based on measurement are shown by points with error flags; the interpolated values are simply joined. The solid point is for liquid H2 ; the open point at 19.2 is for H2 gas. Also shown are the I/Z = 10 ±1 eV band and an early approximation.

=

=

0

:

πε

0 10 20 30 40 50 60 70 80 90 100 8

10

12

14

16

18

20

22

I/Z

(eV

)

Z

Barkas & Berger 1964

ICRU 37 (1984)

0.5

1.0

1.5

2.0

2.5

⟨ – dE

/dx ⟩

min

(MeV

g –1

cm 2

)

1 2 5 10 20 50 100 Z

H He Li Be B C NO Ne Sn Fe

Solids Gases

H 2 gas: 4.10 H 2 liquid: 3.97

2.35 – 0.64 log 10 ( Z ) ##$# 1

& ##

&& BG

'#&$

2D82

#

$#

F 1 4 )8ρ ∗ D82& @#8

*D@7

F 1 4

<

'

$

.

'

;

@ #$#@78

?HH5 −=

8: π=

−−=

#2

δβγββ

!

& 'C8#

Muon momentum

1

10

100

Stop

ping

pow

er [M

eV c

m 2 /

g]

Lin

dhar

d-

Sc

harf

f

Bethe-Bloch

Radiative effects

reach 1%

µ + on Cu

Without δ

Radiative losses

βγ 0.001 0.01 0.1 1 10 100 1000 10 4 10 5 10 6

[MeV/ c ] [GeV/ c ] 100 10 1 0.1 100 10 1 100 10 1

[TeV/ c ]

Anderson- Ziegler

Minimum ionization

E µ c

Nuclear losses

µ −

Current wisdom on Bethe-Bloch

#2

++=

γγβ

D):

D)?

D)

D)

D(

D(

D(

D(?

D(:

D(5

@ #$# @78

#2@7

0.05 0.1 0.02 0.5 0.2 1.0 5.0 2.0 10.0 Pion momentum (GeV/ c )

0.1 0.5 0.2 1.0 5.0 2.0 10.0 50.0 20.0 Proton momentum (GeV/ c )

0.05 0.02 0.1 0.5 0.2 1.0 5.0 2.0 10.0 Muon momentum (GeV/ c )

βγ = p / Mc 1

2

5

10

20

50

100

200

500

1000

2000

5000

10000

20000

50000

R / M

(g c

m − 2

GeV

− 1 )

0.1 2 5 1.0 2 5 10.0 2 5 100.0

H 2 liquid He gas

Pb Fe

C

1

10

100

1000

10000

100000

10 100 1000 10000 100000

Kinetic Energy Proton (MeV)

Range in Iron (g cm-2)

as energy square

as energy

Range of particles in matter.Poor man’s approach:Integrating dE/dX from Rutherford scattering and ignoring the slowly changing ln(term),

" ≈=

Range is approximately proportional to the kinetic energy square at low energy and approximately proportional to the kinetic energy at high energy where the dE/dX is about constant.

Bremsstrahlung and Photon Pair Production.

Radiative Process

e Ze

Zee

Impact parameter : b(non-relativistic!)Peak electric field prop. to e/b2

Characteristic frequency ωc∝1/∆t∝v/2b

IJ:

==⋅

==βα

ωω

ωπω

ω

#

#

$#

IJ

ωβπα

ωωγ ≈

Insert Nγ : photon density Insert the Thomson cross section

# ?

απσ = "

IJ

α

ωασ

ωωσ γ

≈≈

orσB ∼ 0.58mb ∗ Z 2

≈

ωω

σρω

% ρ

≡

%

IJ?

∝=−αα

Radiative Energy Loss.

Define X0 as the Radiative Mean Path.X0 : Radiation Length

B

! 4K8#

∝

? &CCD

&/ & "

[ ]

≈Θ∗=Θ

β

απρ

%

IJ?

∝=−αα

The multiple scattering angle can now be expressed in units of X0

and

Introduce the characteristic energy 5: =⋅≡απ

%

ββ

==Θ

. LK

. # #$#@78

@@$

'

Energy deposit by 1 MeV electrons in 0.53 mm of silicon

The most probable energy loss of an electron of energy 1 MeV in the Si layer is around 200 keV. However, due to the multiple scattering and delta ray production theprimary electron can deposit more energy or even it can be completly absorbed in the detector (in about 4 % of the cases).

http://wwwinfo.cern.ch/asd/geant4/reports/gallery/electromagnetic/edep/summary.html

Electrons of energy 100 MeV have been tracked in aluminium and the longitudinal (z) and tranverse (r)distances travelled by the electrons have been plotted.

B

D@7

Bremsstrahlung

Lead ( Z = 82)

Positrons

Electrons

Ionization Moøller ( e − )

Bhabha ( e + )

Positron annihilation

1.0

0.5

0.20

0.15

0.10

0.05

(cm

2 g − 1

)

E (MeV) 1 0 10 100 1000

1 E

− dE

dx

( X 0 − 1

)

Fractional energy loss per radiation length for electrons and positrons in lead. Critical Energy, Ec , when Bremsstrahlung = Ionization

)::∝

Photon Energy

1 Mb

1 kb

1 b

10 mb10 eV 1 keV 1 MeV 1 GeV 100 GeV

(b) Lead (Z = 82)

σcoherent

σincoh

− experimental σtotσp.e.

σnuc

κN

κe

Cro

ss s

ecti

on (

barn

s/at

om)

Cro

ss s

ecti

on (

barn

s/at

om)

10 mb

1 b

1 kb

1 Mb(a) Carbon (Z = 6)

σcoherent

σincohσnuc

κN

κe

σp.e.

− experimental σtot

Photon total cross sections as a function of energy in carbon andlead, showing the contributions of different processes

σpe= Atomic photo-effect (electron ejection, photon absorption)

σcoherent = Coherent scattering (Rayleigh scattering-atomneither ionised nor excited)

σ incoherent = Incoherent scattering (Compton scattering off an

electron)κn = Pair production, nuclear fieldκe = Pair production, electron fieldσnuc = Photonuclear absorption

(nuclear absorption, usuallyfollowed by emission of a neutron or other particle)

e Ze

Zee

Bremsstrahlung Pair production

5

H ×≈= σσ

Ze

Zee-

e+

! K#

D 98#

W

Pb Sn G10

Fe

Cu

Alglass

Charged Particles in Magnetic Fields.

Dipole Bending Magnet. Quadrupole lens.

Sextupole correction lens.Rare Earth Permanent Magnet. Low β-insertion.

Beam Transport System. Spectrometer Dipole.

α β

φ

φl1 l2

L

R

B

zz

x1 x2x2'

x1'

Dipole Bending Magnet. Rectangular bending magnet. The initial and final displacement and divergence (x1,x1’), (x2,x2’) is defined with respect to the central particle of the beam.(xi’=dxi/dz)It is usual to operate the magnet symmetrical:→ α0=β0=φ/2

( ) ( )&

8?

≅

′

=

′

′

+=

′

φφβα

βα

αβ

kGm/GeV/c

-4

-3

-2

-1

0

1

2

3

4

-00004 -00003 -00002 -00001 00000 00001 00002 00003 00004

NS

N S

R

Quadropole Magnet.

xy=RBx=kyBy=kx

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.5 0 0.5 1 1.5 2

Fiel

d G

radi

ent

z

0 d

k

Assume a simple rectangular model.

=

−=

'("$

)"$

ωωωω

ωω

ωωωωωω

Depending on the plane ( XZ or YZ), the field is either focusing or defocusing.

8

8?

*

+*+ ≅−ω

Thin Lens Approximation.+ no fringe field → d=effective length ≈ pole-length + g*R where g≈1→ drift length * instantaneous change in divergence * drift length

( )

+++

=

−−

(

(

(

(

(

(

(

ωωωωω

ωωω

ωω

+−=

−=

−=

−=

−

−

Focusing

Defocusing

(

→± →

→

→−

ω

ω ω

7 M;F. #

& )& &$ #&$

$ #$/ & ) )

# $& $&& #

& # & # )

Analog Simulation of the Particle Trajectory.Floating Wire.

Take a magnet.Install a (near) mass-less non-magnetic conductive wire.Let the wire pass over a (near) frictionless pulley.Add weight on one end of the wire and fix the other.Add current through the wire.

The central momentum is then given as

p(GeV/c) ≈ 3 10-3 M (g)/i(A)

*!N !'D>'F<#) . #O

@'MF' $ &@'20%$ $MF ' #

##

One can also use:

But floating wire is more fun.

B

' & #%$ $

#'3 *) # $3 * #

$ π(#

Z

Y

X

B

qe,p

αααα

RT

Momentum Measurement and Magnetic Fields.

Solenoidal magnetic field.ALEPH event.WW -> 4 jets

α?

, ≅

α?

, ≅

With B in tesla, momentum in GeV and R in m

or

?

:

5

H

? : 5 H

! 4K

@$ #

!"#$!"#

!"#

SC

+≅

αα

?

,

Momentum measurement can also be done by measuring the multiple scattering.

Y1

Y2

Y3

Y4

Y5

Y6

Dense MaterialX1

X2

X3

X4

X5

High Precision Detector

X6

Θvn

ββββR vt

R

PP'M

Oobserver

Q

b

γτ

τ

τβ

8

=

−=

=

Charged particles do things (particularly if they are moving).

unit vector along τ −

After some manipulation of the 4-vector potential caused by a charge in motion, it can be shown:

[ ]( )

( )

•−

×−×+

•−

−=

×=

βββ

βγβ

?

Velocity field acceleration field

∝

∝

P transverse to the radius vector

It is well known that accelerated charges emit electromagnetic radiation.J. D. JacksonClassical Electrodynamics

Let us assume that the charge is accelerated and the observer is in a frame where the velocity v<<c(we go classic!)

The Lorentz equivalent expression

[ ]

?

?

?

βββγττ

µµ ×−=

−=

-

( ) Θ=××==Ω

=×=

××=

?

:::

::

-

.

πβ

ππ

ππ

β

The energy flux

The radiated power / unit solid angle

The Larmor equation

ββ QQ

That is linear acceleration

→ P ≈ negligible

Circular acceleration

::

?

?

?

?

γβρ

ωγτ

γωτ

-

==→>>=

ρβω =

Energy loss/revolution

5

?

: :

:?

*

-

ργβ

ρπ

βπρ

β−

→ ⋅=∆ →==∆

Take one LEP

* =∆

πρ

H ?⋅= 2 GeV

3.2 10-10 J

0.3 10-5 W/particle

0.3 106 W/bunch

1 eV=1.6 10-19 J

90 µs

∼1011 particles/bunch

ρ

( ) [ ]( ) ( )5

?

885

::

0

Θ−Θ →

•−

×−×=

Ω βπβββ

π ββ

-

The angular distribution of the energy loss for a circular acceleration

Θ→ →

?

:

πγwhich is the Larmorequation (again).

0

100000

200000

0 100000 200000

γγγγ=1.000052∗ 10∗ 10∗ 10∗ 10 5555 ∗∗∗∗ dP(t)/dΩΩΩΩ

γγγγ=2250∗∗∗∗ dP(t)/dΩΩΩΩ

γγγγ=4v

( )( )5

?

0

Θ+Θ≅ →

Ω →Θ γγγ

π

-

γ

≅Θand independent of the

vectorial relationship between ββ P

# $

&#

'$$ #)

& &) #

#$$

# $ & #

$

%$ &#$

/".0$$."

Synchrotron Radiation

Center University

of Wisconsin Madison

Synchrotron radiation spectrum as function of frequency.Circular motion.

ωω

γω

ρω

?

?

=

The critical frequency beyond which there is negligible radiation at any angle:

We will now go on to detectors