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Interaction of light charged particles with matter
Ionization losses ndash electron loss energy as it ionizes and excites atoms
Scattering ndash scattering by Coulomb field of nucleus by field of electrons
Bremsstrahlung radiation ndash enough high energy accelerated motion of charged particle rarr emission of electromagnetic radiation ultrarelativistic energies ndash pair production through virtual photon
Cherenkov radiation ndash charged particle moving faster then light at given material emits electromagnetic radiation in the range of visible lightndash minimal ionization losses
Scattering is induced by interaction with atomic nuclei ( ~ f(Z2) ) and electrons at atomic cloud ( ~ f(Z) ) (difference from heavy particles ndash in this case mainly interactionwith nuclei) energy losses mainly by interaction with electrons at atomic cloud
Motion of electrically charged particles in magnetic and electric fields
Electromagnetic showerndash very high energies
Energy ionization losses
Interaction of electrons ndash interaction of identical particles rarr ΔEMAX = E2
Interaction of positrons ndash they are not identical particles as electrons - anihilation on path end ndash production of 1022 MeV energy
Mostly relativistic harr electrons and positrons are light particles
They will transfer big part of their energy during ionization
Procedure of derivation of equation for ionization losses
1) Classical derivation for nonrelativistic heavy particles2) Quantum derivation for nonrelativistic particles3) Relativistic corrections and corrections on identity of particles for electrons
Ionization losses determination ndash energy losses dx
dE
Bethe - Bloch formulae
Change of momentum
Fdtpb
22
2
04
1
bx
eZF ion
22 bx
bFF
Impact parameter b is changed during scattering only slightly influence of F|| on momentum change are negated (second half negates first)
Influence has only
If velocity v during interaction with one electron changes only slightly transferred momentum is
dx = vdt
Classical derivation (assumption of nonrelativistic velocity and ΔE ltltE )
bv
eZ
bx
x
bv
beZ
bx
dx
v
beZ
v
dxFp ionionion
b
2
0222
2
023
22
2
0
2
4
11
4
1
4
1
Kinetic energy of electron after interaction with ionizing particle
We express path by velocity
22
422
0
2 2
4
1
2 vmb
eZ
m
pE
e
ion
e
eKINe
Constant connected to SI unit system often is putted equal to oneElectric force acts
on particle
bx F||
F FZobrazeniacute siacutelypro elektron
v přiacutepadě iontuje přitažlivaacute
Path of particle passage through matter Δx
Let have thin cylinder (annulus cross-section (bb+db)
Number of electrons at cylinder
Total energy losses at cylinder
Energy losses in the whole roll
eKINecylindr NEE
bdbExndNEE KINeeeKINe 20
where ΔNe ndash number of electrons at cylinder
If charge of material atoms is Z number of electrons ne = Zn0 where n0 ndash atom densityat material We express it by material density ρ Avogardo constant NA and atomic mass A
AeA N
A
Zn
A
Nn
0
and then
bb+db
xbdbndN ee 2
where ne ndash is electron density at material
02
422
0
4
4
1
b
dbn
vm
eZ
dx
dEe
e
ion
02
422
0
4
4
1
b
dbN
A
Z
vm
eZ
dx
dEA
e
ion
22
422
0
2
4
1
vmb
eZE
e
ionKINe
Mention
Limits for integration are not in the reality 0 and infin but bmin and bmax
In the case of integration limits 0 and infin we obtain divergent integral
Maximal energy is transferred during head collision electron obtains energy
22 2)2(2
1)( vmvmMAXEE eeKINeMAX vmpp eMAXeMAX 2because maximal transferred momentum
We use relation between transferred energy and impact parameter
Main dependencyon particle velocity
Main dependency onmaterial properties
Weak dependency on particle velocity and material properties
I
vmN
A
Z
vm
eZ
dx
dE eA
e
ion2
2
422
0
2ln
4
4
1
Constant connected to
SI unit system often is expressed as equal one
2
2
022
422
0
2
4
12
4
12
vm
eZb
vmb
eZvm
e
ionMIN
eMIN
ione
Minimal transferred momentum depends on mean ionization potential of electrons at atom I is and (work achieved through passage must be larger then ionization potential) and corresponding impact parameter is
v
IpMIN
I
eZb
vmb
eZ
vm
I ionMAX
eMAX
ion
e
2
022
422
02
2 2
4
12
4
1
2
2
22
22 vm
I
m
pE
ee
MINMIN
We determine integralMIN
MAXA
e
ion
b
bN
A
Z
vm
eZ
dx
dEln
4
4
12
422
0
where I
vm
vm
eZ
I
eZ
b
b e
e
ion
ion
MIN
MAX2
2
2
0
2
0 2
4
1
2
4
1
and then
Relativistic corrections
2maxmax1
22
mvpmvp ee
In the case of electron rarr identical particles rarr maximal transferred energy ΔEMAX = E2
22
22
22
422
0 1
1ln
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
We obtain early derived equation for v ltlt c
Maximal transferred momentum
Reduction of particle electric field in the direction of flight by factor (1-β2) and in the perpendicular direction increasing by factor
We will obtain on the end
211
This formulae is for electrons even more complex
22222
22
22
22
422
0
118
1)1()112)(2(ln
)1(2ln
2
14
4
1
I
EcmN
A
Z
cm
eZ
dx
dE eA
e
ion
22
22
22
422
0 )1(
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
E ~ up to hundreds MeV rarr light particle losses are 1000 times lower than for heavyE ~ GeV rarr ionization losses of light and heavy particles are comparable
Example of ionization losses for some particles(taken from D Green The physics of particle detector)
Elastic scattering
1) Single scattering
2) Few scatterings
3) Multiple scattering
0
1
nd
0
1~
nd
0
1
nd
Single scattering in the electric field of nucleus ndash described by Rutheford scattering
1) Heavy particles ndash scattering to small angles rarr path is slightly undulated
2) Light particles ndash scattering to large angles rarr range is not defined (for bdquolower energiesldquo)
Mean quadratic deviation from original direction depends on mean quadratic value of scattering angle
(simplified classical derivation for bdquoheavy particlesldquo ndash small scattering angles)
2
204
2cot
ZeZ
bmvπεg
ion
2Θ2
Heavy particles ndash important only for scattering on atomic nuclei
Light particles ndash important also for scattering on electrons
bmvπε
ZeZ ion2
0
2
42tan
2
2220
222
4 bmvπε
ZeZ ion
2min
2max
min
max
2
20
2
2
220
22
220
222
2
2
lnln2
2
1
4
2
2
2
max
min
max
min
max
min
max
min
max
min
max
min
bb
bb
mvπεZeZ
bπ
bmvπε
ZeZπ
bdbπ
dbbmvπε
ZeZπ
πbdb
πbdbb ion
b
b
bb
ion
b
b
b
b
ion
b
b
b
b
2min
2max
max
min
2 bbxA
Nπρbdbπx
A
NρxσNN A
b
b
Aatomroz
22 rozNΘ
min
max22
422
20min
max
2
20
22 ln
2
1ln
2
1
b
b
vp
eZxZ
A
Nρ
πεb
b
mvπε
ZeZxπ
A
NρΘ ionAionA
rarr 0 and then
We determine 2
where Nroz is number of scatterings
2Θ then is determined
Resulting value
1) Strong dependency on momentum2) Strong dependency on velocity 1v4
3) Strong dependency on mass 1m2
4) Strong dependency on particle charge Zion2
5) Strong dependency on material Z Z2
Important scattering properties
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Energy ionization losses
Interaction of electrons ndash interaction of identical particles rarr ΔEMAX = E2
Interaction of positrons ndash they are not identical particles as electrons - anihilation on path end ndash production of 1022 MeV energy
Mostly relativistic harr electrons and positrons are light particles
They will transfer big part of their energy during ionization
Procedure of derivation of equation for ionization losses
1) Classical derivation for nonrelativistic heavy particles2) Quantum derivation for nonrelativistic particles3) Relativistic corrections and corrections on identity of particles for electrons
Ionization losses determination ndash energy losses dx
dE
Bethe - Bloch formulae
Change of momentum
Fdtpb
22
2
04
1
bx
eZF ion
22 bx
bFF
Impact parameter b is changed during scattering only slightly influence of F|| on momentum change are negated (second half negates first)
Influence has only
If velocity v during interaction with one electron changes only slightly transferred momentum is
dx = vdt
Classical derivation (assumption of nonrelativistic velocity and ΔE ltltE )
bv
eZ
bx
x
bv
beZ
bx
dx
v
beZ
v
dxFp ionionion
b
2
0222
2
023
22
2
0
2
4
11
4
1
4
1
Kinetic energy of electron after interaction with ionizing particle
We express path by velocity
22
422
0
2 2
4
1
2 vmb
eZ
m
pE
e
ion
e
eKINe
Constant connected to SI unit system often is putted equal to oneElectric force acts
on particle
bx F||
F FZobrazeniacute siacutelypro elektron
v přiacutepadě iontuje přitažlivaacute
Path of particle passage through matter Δx
Let have thin cylinder (annulus cross-section (bb+db)
Number of electrons at cylinder
Total energy losses at cylinder
Energy losses in the whole roll
eKINecylindr NEE
bdbExndNEE KINeeeKINe 20
where ΔNe ndash number of electrons at cylinder
If charge of material atoms is Z number of electrons ne = Zn0 where n0 ndash atom densityat material We express it by material density ρ Avogardo constant NA and atomic mass A
AeA N
A
Zn
A
Nn
0
and then
bb+db
xbdbndN ee 2
where ne ndash is electron density at material
02
422
0
4
4
1
b
dbn
vm
eZ
dx
dEe
e
ion
02
422
0
4
4
1
b
dbN
A
Z
vm
eZ
dx
dEA
e
ion
22
422
0
2
4
1
vmb
eZE
e
ionKINe
Mention
Limits for integration are not in the reality 0 and infin but bmin and bmax
In the case of integration limits 0 and infin we obtain divergent integral
Maximal energy is transferred during head collision electron obtains energy
22 2)2(2
1)( vmvmMAXEE eeKINeMAX vmpp eMAXeMAX 2because maximal transferred momentum
We use relation between transferred energy and impact parameter
Main dependencyon particle velocity
Main dependency onmaterial properties
Weak dependency on particle velocity and material properties
I
vmN
A
Z
vm
eZ
dx
dE eA
e
ion2
2
422
0
2ln
4
4
1
Constant connected to
SI unit system often is expressed as equal one
2
2
022
422
0
2
4
12
4
12
vm
eZb
vmb
eZvm
e
ionMIN
eMIN
ione
Minimal transferred momentum depends on mean ionization potential of electrons at atom I is and (work achieved through passage must be larger then ionization potential) and corresponding impact parameter is
v
IpMIN
I
eZb
vmb
eZ
vm
I ionMAX
eMAX
ion
e
2
022
422
02
2 2
4
12
4
1
2
2
22
22 vm
I
m
pE
ee
MINMIN
We determine integralMIN
MAXA
e
ion
b
bN
A
Z
vm
eZ
dx
dEln
4
4
12
422
0
where I
vm
vm
eZ
I
eZ
b
b e
e
ion
ion
MIN
MAX2
2
2
0
2
0 2
4
1
2
4
1
and then
Relativistic corrections
2maxmax1
22
mvpmvp ee
In the case of electron rarr identical particles rarr maximal transferred energy ΔEMAX = E2
22
22
22
422
0 1
1ln
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
We obtain early derived equation for v ltlt c
Maximal transferred momentum
Reduction of particle electric field in the direction of flight by factor (1-β2) and in the perpendicular direction increasing by factor
We will obtain on the end
211
This formulae is for electrons even more complex
22222
22
22
22
422
0
118
1)1()112)(2(ln
)1(2ln
2
14
4
1
I
EcmN
A
Z
cm
eZ
dx
dE eA
e
ion
22
22
22
422
0 )1(
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
E ~ up to hundreds MeV rarr light particle losses are 1000 times lower than for heavyE ~ GeV rarr ionization losses of light and heavy particles are comparable
Example of ionization losses for some particles(taken from D Green The physics of particle detector)
Elastic scattering
1) Single scattering
2) Few scatterings
3) Multiple scattering
0
1
nd
0
1~
nd
0
1
nd
Single scattering in the electric field of nucleus ndash described by Rutheford scattering
1) Heavy particles ndash scattering to small angles rarr path is slightly undulated
2) Light particles ndash scattering to large angles rarr range is not defined (for bdquolower energiesldquo)
Mean quadratic deviation from original direction depends on mean quadratic value of scattering angle
(simplified classical derivation for bdquoheavy particlesldquo ndash small scattering angles)
2
204
2cot
ZeZ
bmvπεg
ion
2Θ2
Heavy particles ndash important only for scattering on atomic nuclei
Light particles ndash important also for scattering on electrons
bmvπε
ZeZ ion2
0
2
42tan
2
2220
222
4 bmvπε
ZeZ ion
2min
2max
min
max
2
20
2
2
220
22
220
222
2
2
lnln2
2
1
4
2
2
2
max
min
max
min
max
min
max
min
max
min
max
min
bb
bb
mvπεZeZ
bπ
bmvπε
ZeZπ
bdbπ
dbbmvπε
ZeZπ
πbdb
πbdbb ion
b
b
bb
ion
b
b
b
b
ion
b
b
b
b
2min
2max
max
min
2 bbxA
Nπρbdbπx
A
NρxσNN A
b
b
Aatomroz
22 rozNΘ
min
max22
422
20min
max
2
20
22 ln
2
1ln
2
1
b
b
vp
eZxZ
A
Nρ
πεb
b
mvπε
ZeZxπ
A
NρΘ ionAionA
rarr 0 and then
We determine 2
where Nroz is number of scatterings
2Θ then is determined
Resulting value
1) Strong dependency on momentum2) Strong dependency on velocity 1v4
3) Strong dependency on mass 1m2
4) Strong dependency on particle charge Zion2
5) Strong dependency on material Z Z2
Important scattering properties
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Bethe - Bloch formulae
Change of momentum
Fdtpb
22
2
04
1
bx
eZF ion
22 bx
bFF
Impact parameter b is changed during scattering only slightly influence of F|| on momentum change are negated (second half negates first)
Influence has only
If velocity v during interaction with one electron changes only slightly transferred momentum is
dx = vdt
Classical derivation (assumption of nonrelativistic velocity and ΔE ltltE )
bv
eZ
bx
x
bv
beZ
bx
dx
v
beZ
v
dxFp ionionion
b
2
0222
2
023
22
2
0
2
4
11
4
1
4
1
Kinetic energy of electron after interaction with ionizing particle
We express path by velocity
22
422
0
2 2
4
1
2 vmb
eZ
m
pE
e
ion
e
eKINe
Constant connected to SI unit system often is putted equal to oneElectric force acts
on particle
bx F||
F FZobrazeniacute siacutelypro elektron
v přiacutepadě iontuje přitažlivaacute
Path of particle passage through matter Δx
Let have thin cylinder (annulus cross-section (bb+db)
Number of electrons at cylinder
Total energy losses at cylinder
Energy losses in the whole roll
eKINecylindr NEE
bdbExndNEE KINeeeKINe 20
where ΔNe ndash number of electrons at cylinder
If charge of material atoms is Z number of electrons ne = Zn0 where n0 ndash atom densityat material We express it by material density ρ Avogardo constant NA and atomic mass A
AeA N
A
Zn
A
Nn
0
and then
bb+db
xbdbndN ee 2
where ne ndash is electron density at material
02
422
0
4
4
1
b
dbn
vm
eZ
dx
dEe
e
ion
02
422
0
4
4
1
b
dbN
A
Z
vm
eZ
dx
dEA
e
ion
22
422
0
2
4
1
vmb
eZE
e
ionKINe
Mention
Limits for integration are not in the reality 0 and infin but bmin and bmax
In the case of integration limits 0 and infin we obtain divergent integral
Maximal energy is transferred during head collision electron obtains energy
22 2)2(2
1)( vmvmMAXEE eeKINeMAX vmpp eMAXeMAX 2because maximal transferred momentum
We use relation between transferred energy and impact parameter
Main dependencyon particle velocity
Main dependency onmaterial properties
Weak dependency on particle velocity and material properties
I
vmN
A
Z
vm
eZ
dx
dE eA
e
ion2
2
422
0
2ln
4
4
1
Constant connected to
SI unit system often is expressed as equal one
2
2
022
422
0
2
4
12
4
12
vm
eZb
vmb
eZvm
e
ionMIN
eMIN
ione
Minimal transferred momentum depends on mean ionization potential of electrons at atom I is and (work achieved through passage must be larger then ionization potential) and corresponding impact parameter is
v
IpMIN
I
eZb
vmb
eZ
vm
I ionMAX
eMAX
ion
e
2
022
422
02
2 2
4
12
4
1
2
2
22
22 vm
I
m
pE
ee
MINMIN
We determine integralMIN
MAXA
e
ion
b
bN
A
Z
vm
eZ
dx
dEln
4
4
12
422
0
where I
vm
vm
eZ
I
eZ
b
b e
e
ion
ion
MIN
MAX2
2
2
0
2
0 2
4
1
2
4
1
and then
Relativistic corrections
2maxmax1
22
mvpmvp ee
In the case of electron rarr identical particles rarr maximal transferred energy ΔEMAX = E2
22
22
22
422
0 1
1ln
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
We obtain early derived equation for v ltlt c
Maximal transferred momentum
Reduction of particle electric field in the direction of flight by factor (1-β2) and in the perpendicular direction increasing by factor
We will obtain on the end
211
This formulae is for electrons even more complex
22222
22
22
22
422
0
118
1)1()112)(2(ln
)1(2ln
2
14
4
1
I
EcmN
A
Z
cm
eZ
dx
dE eA
e
ion
22
22
22
422
0 )1(
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
E ~ up to hundreds MeV rarr light particle losses are 1000 times lower than for heavyE ~ GeV rarr ionization losses of light and heavy particles are comparable
Example of ionization losses for some particles(taken from D Green The physics of particle detector)
Elastic scattering
1) Single scattering
2) Few scatterings
3) Multiple scattering
0
1
nd
0
1~
nd
0
1
nd
Single scattering in the electric field of nucleus ndash described by Rutheford scattering
1) Heavy particles ndash scattering to small angles rarr path is slightly undulated
2) Light particles ndash scattering to large angles rarr range is not defined (for bdquolower energiesldquo)
Mean quadratic deviation from original direction depends on mean quadratic value of scattering angle
(simplified classical derivation for bdquoheavy particlesldquo ndash small scattering angles)
2
204
2cot
ZeZ
bmvπεg
ion
2Θ2
Heavy particles ndash important only for scattering on atomic nuclei
Light particles ndash important also for scattering on electrons
bmvπε
ZeZ ion2
0
2
42tan
2
2220
222
4 bmvπε
ZeZ ion
2min
2max
min
max
2
20
2
2
220
22
220
222
2
2
lnln2
2
1
4
2
2
2
max
min
max
min
max
min
max
min
max
min
max
min
bb
bb
mvπεZeZ
bπ
bmvπε
ZeZπ
bdbπ
dbbmvπε
ZeZπ
πbdb
πbdbb ion
b
b
bb
ion
b
b
b
b
ion
b
b
b
b
2min
2max
max
min
2 bbxA
Nπρbdbπx
A
NρxσNN A
b
b
Aatomroz
22 rozNΘ
min
max22
422
20min
max
2
20
22 ln
2
1ln
2
1
b
b
vp
eZxZ
A
Nρ
πεb
b
mvπε
ZeZxπ
A
NρΘ ionAionA
rarr 0 and then
We determine 2
where Nroz is number of scatterings
2Θ then is determined
Resulting value
1) Strong dependency on momentum2) Strong dependency on velocity 1v4
3) Strong dependency on mass 1m2
4) Strong dependency on particle charge Zion2
5) Strong dependency on material Z Z2
Important scattering properties
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Path of particle passage through matter Δx
Let have thin cylinder (annulus cross-section (bb+db)
Number of electrons at cylinder
Total energy losses at cylinder
Energy losses in the whole roll
eKINecylindr NEE
bdbExndNEE KINeeeKINe 20
where ΔNe ndash number of electrons at cylinder
If charge of material atoms is Z number of electrons ne = Zn0 where n0 ndash atom densityat material We express it by material density ρ Avogardo constant NA and atomic mass A
AeA N
A
Zn
A
Nn
0
and then
bb+db
xbdbndN ee 2
where ne ndash is electron density at material
02
422
0
4
4
1
b
dbn
vm
eZ
dx
dEe
e
ion
02
422
0
4
4
1
b
dbN
A
Z
vm
eZ
dx
dEA
e
ion
22
422
0
2
4
1
vmb
eZE
e
ionKINe
Mention
Limits for integration are not in the reality 0 and infin but bmin and bmax
In the case of integration limits 0 and infin we obtain divergent integral
Maximal energy is transferred during head collision electron obtains energy
22 2)2(2
1)( vmvmMAXEE eeKINeMAX vmpp eMAXeMAX 2because maximal transferred momentum
We use relation between transferred energy and impact parameter
Main dependencyon particle velocity
Main dependency onmaterial properties
Weak dependency on particle velocity and material properties
I
vmN
A
Z
vm
eZ
dx
dE eA
e
ion2
2
422
0
2ln
4
4
1
Constant connected to
SI unit system often is expressed as equal one
2
2
022
422
0
2
4
12
4
12
vm
eZb
vmb
eZvm
e
ionMIN
eMIN
ione
Minimal transferred momentum depends on mean ionization potential of electrons at atom I is and (work achieved through passage must be larger then ionization potential) and corresponding impact parameter is
v
IpMIN
I
eZb
vmb
eZ
vm
I ionMAX
eMAX
ion
e
2
022
422
02
2 2
4
12
4
1
2
2
22
22 vm
I
m
pE
ee
MINMIN
We determine integralMIN
MAXA
e
ion
b
bN
A
Z
vm
eZ
dx
dEln
4
4
12
422
0
where I
vm
vm
eZ
I
eZ
b
b e
e
ion
ion
MIN
MAX2
2
2
0
2
0 2
4
1
2
4
1
and then
Relativistic corrections
2maxmax1
22
mvpmvp ee
In the case of electron rarr identical particles rarr maximal transferred energy ΔEMAX = E2
22
22
22
422
0 1
1ln
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
We obtain early derived equation for v ltlt c
Maximal transferred momentum
Reduction of particle electric field in the direction of flight by factor (1-β2) and in the perpendicular direction increasing by factor
We will obtain on the end
211
This formulae is for electrons even more complex
22222
22
22
22
422
0
118
1)1()112)(2(ln
)1(2ln
2
14
4
1
I
EcmN
A
Z
cm
eZ
dx
dE eA
e
ion
22
22
22
422
0 )1(
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
E ~ up to hundreds MeV rarr light particle losses are 1000 times lower than for heavyE ~ GeV rarr ionization losses of light and heavy particles are comparable
Example of ionization losses for some particles(taken from D Green The physics of particle detector)
Elastic scattering
1) Single scattering
2) Few scatterings
3) Multiple scattering
0
1
nd
0
1~
nd
0
1
nd
Single scattering in the electric field of nucleus ndash described by Rutheford scattering
1) Heavy particles ndash scattering to small angles rarr path is slightly undulated
2) Light particles ndash scattering to large angles rarr range is not defined (for bdquolower energiesldquo)
Mean quadratic deviation from original direction depends on mean quadratic value of scattering angle
(simplified classical derivation for bdquoheavy particlesldquo ndash small scattering angles)
2
204
2cot
ZeZ
bmvπεg
ion
2Θ2
Heavy particles ndash important only for scattering on atomic nuclei
Light particles ndash important also for scattering on electrons
bmvπε
ZeZ ion2
0
2
42tan
2
2220
222
4 bmvπε
ZeZ ion
2min
2max
min
max
2
20
2
2
220
22
220
222
2
2
lnln2
2
1
4
2
2
2
max
min
max
min
max
min
max
min
max
min
max
min
bb
bb
mvπεZeZ
bπ
bmvπε
ZeZπ
bdbπ
dbbmvπε
ZeZπ
πbdb
πbdbb ion
b
b
bb
ion
b
b
b
b
ion
b
b
b
b
2min
2max
max
min
2 bbxA
Nπρbdbπx
A
NρxσNN A
b
b
Aatomroz
22 rozNΘ
min
max22
422
20min
max
2
20
22 ln
2
1ln
2
1
b
b
vp
eZxZ
A
Nρ
πεb
b
mvπε
ZeZxπ
A
NρΘ ionAionA
rarr 0 and then
We determine 2
where Nroz is number of scatterings
2Θ then is determined
Resulting value
1) Strong dependency on momentum2) Strong dependency on velocity 1v4
3) Strong dependency on mass 1m2
4) Strong dependency on particle charge Zion2
5) Strong dependency on material Z Z2
Important scattering properties
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Limits for integration are not in the reality 0 and infin but bmin and bmax
In the case of integration limits 0 and infin we obtain divergent integral
Maximal energy is transferred during head collision electron obtains energy
22 2)2(2
1)( vmvmMAXEE eeKINeMAX vmpp eMAXeMAX 2because maximal transferred momentum
We use relation between transferred energy and impact parameter
Main dependencyon particle velocity
Main dependency onmaterial properties
Weak dependency on particle velocity and material properties
I
vmN
A
Z
vm
eZ
dx
dE eA
e
ion2
2
422
0
2ln
4
4
1
Constant connected to
SI unit system often is expressed as equal one
2
2
022
422
0
2
4
12
4
12
vm
eZb
vmb
eZvm
e
ionMIN
eMIN
ione
Minimal transferred momentum depends on mean ionization potential of electrons at atom I is and (work achieved through passage must be larger then ionization potential) and corresponding impact parameter is
v
IpMIN
I
eZb
vmb
eZ
vm
I ionMAX
eMAX
ion
e
2
022
422
02
2 2
4
12
4
1
2
2
22
22 vm
I
m
pE
ee
MINMIN
We determine integralMIN
MAXA
e
ion
b
bN
A
Z
vm
eZ
dx
dEln
4
4
12
422
0
where I
vm
vm
eZ
I
eZ
b
b e
e
ion
ion
MIN
MAX2
2
2
0
2
0 2
4
1
2
4
1
and then
Relativistic corrections
2maxmax1
22
mvpmvp ee
In the case of electron rarr identical particles rarr maximal transferred energy ΔEMAX = E2
22
22
22
422
0 1
1ln
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
We obtain early derived equation for v ltlt c
Maximal transferred momentum
Reduction of particle electric field in the direction of flight by factor (1-β2) and in the perpendicular direction increasing by factor
We will obtain on the end
211
This formulae is for electrons even more complex
22222
22
22
22
422
0
118
1)1()112)(2(ln
)1(2ln
2
14
4
1
I
EcmN
A
Z
cm
eZ
dx
dE eA
e
ion
22
22
22
422
0 )1(
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
E ~ up to hundreds MeV rarr light particle losses are 1000 times lower than for heavyE ~ GeV rarr ionization losses of light and heavy particles are comparable
Example of ionization losses for some particles(taken from D Green The physics of particle detector)
Elastic scattering
1) Single scattering
2) Few scatterings
3) Multiple scattering
0
1
nd
0
1~
nd
0
1
nd
Single scattering in the electric field of nucleus ndash described by Rutheford scattering
1) Heavy particles ndash scattering to small angles rarr path is slightly undulated
2) Light particles ndash scattering to large angles rarr range is not defined (for bdquolower energiesldquo)
Mean quadratic deviation from original direction depends on mean quadratic value of scattering angle
(simplified classical derivation for bdquoheavy particlesldquo ndash small scattering angles)
2
204
2cot
ZeZ
bmvπεg
ion
2Θ2
Heavy particles ndash important only for scattering on atomic nuclei
Light particles ndash important also for scattering on electrons
bmvπε
ZeZ ion2
0
2
42tan
2
2220
222
4 bmvπε
ZeZ ion
2min
2max
min
max
2
20
2
2
220
22
220
222
2
2
lnln2
2
1
4
2
2
2
max
min
max
min
max
min
max
min
max
min
max
min
bb
bb
mvπεZeZ
bπ
bmvπε
ZeZπ
bdbπ
dbbmvπε
ZeZπ
πbdb
πbdbb ion
b
b
bb
ion
b
b
b
b
ion
b
b
b
b
2min
2max
max
min
2 bbxA
Nπρbdbπx
A
NρxσNN A
b
b
Aatomroz
22 rozNΘ
min
max22
422
20min
max
2
20
22 ln
2
1ln
2
1
b
b
vp
eZxZ
A
Nρ
πεb
b
mvπε
ZeZxπ
A
NρΘ ionAionA
rarr 0 and then
We determine 2
where Nroz is number of scatterings
2Θ then is determined
Resulting value
1) Strong dependency on momentum2) Strong dependency on velocity 1v4
3) Strong dependency on mass 1m2
4) Strong dependency on particle charge Zion2
5) Strong dependency on material Z Z2
Important scattering properties
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Relativistic corrections
2maxmax1
22
mvpmvp ee
In the case of electron rarr identical particles rarr maximal transferred energy ΔEMAX = E2
22
22
22
422
0 1
1ln
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
We obtain early derived equation for v ltlt c
Maximal transferred momentum
Reduction of particle electric field in the direction of flight by factor (1-β2) and in the perpendicular direction increasing by factor
We will obtain on the end
211
This formulae is for electrons even more complex
22222
22
22
22
422
0
118
1)1()112)(2(ln
)1(2ln
2
14
4
1
I
EcmN
A
Z
cm
eZ
dx
dE eA
e
ion
22
22
22
422
0 )1(
2ln
4
4
1
I
cmN
A
Z
cm
eZ
dx
dE eA
e
ion
E ~ up to hundreds MeV rarr light particle losses are 1000 times lower than for heavyE ~ GeV rarr ionization losses of light and heavy particles are comparable
Example of ionization losses for some particles(taken from D Green The physics of particle detector)
Elastic scattering
1) Single scattering
2) Few scatterings
3) Multiple scattering
0
1
nd
0
1~
nd
0
1
nd
Single scattering in the electric field of nucleus ndash described by Rutheford scattering
1) Heavy particles ndash scattering to small angles rarr path is slightly undulated
2) Light particles ndash scattering to large angles rarr range is not defined (for bdquolower energiesldquo)
Mean quadratic deviation from original direction depends on mean quadratic value of scattering angle
(simplified classical derivation for bdquoheavy particlesldquo ndash small scattering angles)
2
204
2cot
ZeZ
bmvπεg
ion
2Θ2
Heavy particles ndash important only for scattering on atomic nuclei
Light particles ndash important also for scattering on electrons
bmvπε
ZeZ ion2
0
2
42tan
2
2220
222
4 bmvπε
ZeZ ion
2min
2max
min
max
2
20
2
2
220
22
220
222
2
2
lnln2
2
1
4
2
2
2
max
min
max
min
max
min
max
min
max
min
max
min
bb
bb
mvπεZeZ
bπ
bmvπε
ZeZπ
bdbπ
dbbmvπε
ZeZπ
πbdb
πbdbb ion
b
b
bb
ion
b
b
b
b
ion
b
b
b
b
2min
2max
max
min
2 bbxA
Nπρbdbπx
A
NρxσNN A
b
b
Aatomroz
22 rozNΘ
min
max22
422
20min
max
2
20
22 ln
2
1ln
2
1
b
b
vp
eZxZ
A
Nρ
πεb
b
mvπε
ZeZxπ
A
NρΘ ionAionA
rarr 0 and then
We determine 2
where Nroz is number of scatterings
2Θ then is determined
Resulting value
1) Strong dependency on momentum2) Strong dependency on velocity 1v4
3) Strong dependency on mass 1m2
4) Strong dependency on particle charge Zion2
5) Strong dependency on material Z Z2
Important scattering properties
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Example of ionization losses for some particles(taken from D Green The physics of particle detector)
Elastic scattering
1) Single scattering
2) Few scatterings
3) Multiple scattering
0
1
nd
0
1~
nd
0
1
nd
Single scattering in the electric field of nucleus ndash described by Rutheford scattering
1) Heavy particles ndash scattering to small angles rarr path is slightly undulated
2) Light particles ndash scattering to large angles rarr range is not defined (for bdquolower energiesldquo)
Mean quadratic deviation from original direction depends on mean quadratic value of scattering angle
(simplified classical derivation for bdquoheavy particlesldquo ndash small scattering angles)
2
204
2cot
ZeZ
bmvπεg
ion
2Θ2
Heavy particles ndash important only for scattering on atomic nuclei
Light particles ndash important also for scattering on electrons
bmvπε
ZeZ ion2
0
2
42tan
2
2220
222
4 bmvπε
ZeZ ion
2min
2max
min
max
2
20
2
2
220
22
220
222
2
2
lnln2
2
1
4
2
2
2
max
min
max
min
max
min
max
min
max
min
max
min
bb
bb
mvπεZeZ
bπ
bmvπε
ZeZπ
bdbπ
dbbmvπε
ZeZπ
πbdb
πbdbb ion
b
b
bb
ion
b
b
b
b
ion
b
b
b
b
2min
2max
max
min
2 bbxA
Nπρbdbπx
A
NρxσNN A
b
b
Aatomroz
22 rozNΘ
min
max22
422
20min
max
2
20
22 ln
2
1ln
2
1
b
b
vp
eZxZ
A
Nρ
πεb
b
mvπε
ZeZxπ
A
NρΘ ionAionA
rarr 0 and then
We determine 2
where Nroz is number of scatterings
2Θ then is determined
Resulting value
1) Strong dependency on momentum2) Strong dependency on velocity 1v4
3) Strong dependency on mass 1m2
4) Strong dependency on particle charge Zion2
5) Strong dependency on material Z Z2
Important scattering properties
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Elastic scattering
1) Single scattering
2) Few scatterings
3) Multiple scattering
0
1
nd
0
1~
nd
0
1
nd
Single scattering in the electric field of nucleus ndash described by Rutheford scattering
1) Heavy particles ndash scattering to small angles rarr path is slightly undulated
2) Light particles ndash scattering to large angles rarr range is not defined (for bdquolower energiesldquo)
Mean quadratic deviation from original direction depends on mean quadratic value of scattering angle
(simplified classical derivation for bdquoheavy particlesldquo ndash small scattering angles)
2
204
2cot
ZeZ
bmvπεg
ion
2Θ2
Heavy particles ndash important only for scattering on atomic nuclei
Light particles ndash important also for scattering on electrons
bmvπε
ZeZ ion2
0
2
42tan
2
2220
222
4 bmvπε
ZeZ ion
2min
2max
min
max
2
20
2
2
220
22
220
222
2
2
lnln2
2
1
4
2
2
2
max
min
max
min
max
min
max
min
max
min
max
min
bb
bb
mvπεZeZ
bπ
bmvπε
ZeZπ
bdbπ
dbbmvπε
ZeZπ
πbdb
πbdbb ion
b
b
bb
ion
b
b
b
b
ion
b
b
b
b
2min
2max
max
min
2 bbxA
Nπρbdbπx
A
NρxσNN A
b
b
Aatomroz
22 rozNΘ
min
max22
422
20min
max
2
20
22 ln
2
1ln
2
1
b
b
vp
eZxZ
A
Nρ
πεb
b
mvπε
ZeZxπ
A
NρΘ ionAionA
rarr 0 and then
We determine 2
where Nroz is number of scatterings
2Θ then is determined
Resulting value
1) Strong dependency on momentum2) Strong dependency on velocity 1v4
3) Strong dependency on mass 1m2
4) Strong dependency on particle charge Zion2
5) Strong dependency on material Z Z2
Important scattering properties
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
bmvπε
ZeZ ion2
0
2
42tan
2
2220
222
4 bmvπε
ZeZ ion
2min
2max
min
max
2
20
2
2
220
22
220
222
2
2
lnln2
2
1
4
2
2
2
max
min
max
min
max
min
max
min
max
min
max
min
bb
bb
mvπεZeZ
bπ
bmvπε
ZeZπ
bdbπ
dbbmvπε
ZeZπ
πbdb
πbdbb ion
b
b
bb
ion
b
b
b
b
ion
b
b
b
b
2min
2max
max
min
2 bbxA
Nπρbdbπx
A
NρxσNN A
b
b
Aatomroz
22 rozNΘ
min
max22
422
20min
max
2
20
22 ln
2
1ln
2
1
b
b
vp
eZxZ
A
Nρ
πεb
b
mvπε
ZeZxπ
A
NρΘ ionAionA
rarr 0 and then
We determine 2
where Nroz is number of scatterings
2Θ then is determined
Resulting value
1) Strong dependency on momentum2) Strong dependency on velocity 1v4
3) Strong dependency on mass 1m2
4) Strong dependency on particle charge Zion2
5) Strong dependency on material Z Z2
Important scattering properties
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation
Energy emitted per time unit 2~ adt
dE
m
1
r
ZeZ
4
1
m
Fa
2
2ion
0
C
Acceleration is given by Coulomb interaction
For proton and electron6
2
2
2
2
2
1030938
51101
1
)(
)(
MeV
MeV
m
m
m
m
elektrondt
dE
protondt
dE
p
e
e
p
rad
rad
For muon and electron is same ratio 2610-5
Radiation losses show itself in bdquonormal situationldquo only for electrons and positronsFor ultrarelativistic energies also for further particles
Dependency on material charge ion charge
and mass
2~ ionZdt
dE2~ Z
dt
dE
2
1~
mdt
dE
Rozdiacutel v naacuteboji iontu malyacute v hmotnosti mnohem většiacute
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
)(4 220 ZEFZEr
A
N
dx
dE A
rad
Course of function F(EZ) depends on energy (E0 ndash initial electron energy)and if it is necessary count screening of electrons
Without screening
312
02 Zcm
Ecm ee
Complete screening
312
0
ZcmE e
3
1)(2ln)( ZfZEF
18
1)()183ln()( 31 ZfZZEF
20
cm
E
e
where
(it is similar calculation and result as for pair production ndash see gamma ray interaction)
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1
Description is equivalent pair production description
mcm
er
e
152
0
2
0 108224
137
1
4 0
2
c
e
where for mention
and F(EZ) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E
E asymp hν0 ndash eigenfrequency of atom rarr interaction with atom ndash screening has not influence
E gtgt hν0 ndash interaction with nucleus rarr screening is necessary count according to where electron interacts with nucleus
Low energy rarr strong field near nucleus is necessary
High energy rarr weak field further from nucleus is enough ndash there is maximum of production
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
For radiation length
Critical energy ECionrad
C dx
dE
dx
dEEE
Radiation losses linearly proportional to energy EXdx
dE
rad 0
1
)(41 22
00
ZEFZrA
N
XA
00)( X
x
eExE
Energy losses of electron (if they are only radiation losses)
For electron and positron is EC gt mec2 rarr v asymp c
)(4
4
1)(4
2
42
0
220 EFN
A
Z
cm
eZEFZEr
A
N
dx
dE
dx
dEionA
erad
A
ionrad
)(
)(
)(
)(2 EF
ZEFZ
EF
ZEFZ
cm
E
dx
dE
dx
dE
ion
rad
ion
rad
e
ion
rad
)(
)(12 ZEF
EF
Zcm
E
rad
ion
e
CC
EC [MeV]Air 80Al 40Pb 76
for v rarr c is valid Fion(E) = f(lnE) Let approve
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Total energy losses
Total losses are given by ionization and radiation losses
ri dx
dE
dx
dE
dx
dE
Electron range absorption Protons Electrons
Schematic comparison of different quantities for protons and electrons
Well defined range does not exist
Rextrap - extrapolated path ndash point fo linear extrapolation crossing
We obtain exponential dependency for spectrum of beta emitter
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons(taken from D Green The physics of particle detector)
Ultrarelativistic energies
Electromagnetic shower creation ndash see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Angular and energy distributions of bremsstrahlung photons
Depends on electron (other particle) energy does not depend on emitted photon energy
Mean angle of photon emission TOT
2e
S E
cmΘ~ Erarr infin ΘS rarr 0
Photons are emitted to narrow cone to the direction of electron motion preference of forward angles increases with energy
Angular distribution
Energy distribution
Maximal possible emitted energy ndash kinetic energy of electron
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Synchrotron radiationSimilar origin as bremsstrahlung ndash it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons) Influence of acceleration rarr emission of electromagnetic radiation
Synchrotron radiation is not connected with material ndash lower acceleration rarr it has lower energy
R
vγ
dt
dvγ
dt
)vd(
m
γ
td
dp
m
1a
222
γm
Acting force is Lorentz force BvqFL
2
3
2
04
1
3
2a
c
Ze
dt
dE
Energy losses
Classical centripetal acceleration a=v2R
2
4
3
2
04
1
3
2
R
v
c
Ze
dt
dE
442
2
02
44
3
2
0
222
3
2
0
γcR4
1
3
2
R
vγ
4
1
3
2
R
vγ
4
1
3
2
Ze
c
Ze
c
Ze
dt
dE
Relativistic centripetal acceleration
Energy losses
Energy losses
22
1
1γ
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Cherenkov radiation
Particle velocity in the material v gt crsquo = cn (n ndash index of refraction) rarr emission of Cherenkov radiation
nv
c
vt
tnc
cos n
1cos
Results of this equation
1) Threshold velocity exists βmin = 1n For βmin emission is in the direction of particle motion Cherenkov radiation is not produced for lower velocities
2) For ultrarelativistic particles cos Θmax = 1n
3) For water n = 133 rarr βmin = 075 for electron EKIN = 026 MeV cosΘmax = 075 rarr Θmax= 415o
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
Transition radiation
Passage of charged particle through boundary of materials with different index of refraction rarr emission of electromagnetic radiation (discovery of Ginsburg Frank 1946)
e-
+
+
vacuummaterialCreation of dipole in boundary zone rarr dipole elmg field changes in time rarr emission of elmg radiation
Energy emitted by one transition materialvacuum
Number of photons emitted on boundary (is very small necessity of many transitions)
~3
1E P
High energy electron emits transition radiation
~Nf E
plasma frequency ħωP asymp 14 eV (for Li) 07 eV (air) 20 eV (for polyethylene)
Emission sharply directed to the particle flight direction1
~
Radiators of transition radiation material with small Z reabsorption increases with ~ Z5
Energy of emitted photons 10 ndash 30 keV 0000020~20000
20
137
1
3
1~Nf eV
eV
Good combination of radiators and X-ray detectors
