PHYS 352

Energy Loss by Electrons

Passage of Charged Particles through Matter

• two things happen: a) the particle loses energy traversing matter and b) particle is deflected from its initial direction

• two main processes cause this: 1) inelastic collisions with atomic electrons in the material and 2) elastic scattering off nuclei

• other processes also cause energy loss: 3) bremsstrahlung, 4) emission of Cherenkov radiation (relative of bremsstrahlung), 5) nuclear reactions (rare, lower probability)

• makes sense to separate the consideration of heavy charged particles and light charged particles (i.e. electrons)

• heavy particles don’t undergo 3) and 4); 5) is rare; 2) is again less common compared to 1)...and heavy particles don’t deflect much off electrons

• basically need only consider inelastic collision with atomic electrons for energy loss of heavy charged particles

Differences between Electrons and Heavy Charged Particles

• electrons are light and collide with other (atomic) electrons

• assumption that they are undeflected as they plow through matter is invalid (i.e. large angle multiple scattering does occur)

• assumption that each energy loss event is a small fraction of the incident energy is invalid (large energy loss possible electrons off electrons)

• scattering off identical particles; must take into account indistinguishability (in the quantum sense)

• electrons are definitely relativistic, at nuclear energies

• electrons emit radiation as they lose energy, contributing to the energy loss

• bremsstrahlung

• Cherenkov radiation

Collision Energy Loss for Electrons

• modify the normal Bethe-Bloch formula

• and it becomes, for electrons

• where τ=γ–1 is equal to the kinetic energy of the electron divided by mec2

• and the function F(τ):

• why are electrons and positrons different?

• what’s the maximum energy transfer: electron colliding with an electron?

• what’s the maximum energy transfer: positron colliding with an electron?

−dEdx

=4π k2e4

mec2

z2

β 2

ρZ NA

AB(v); B(v) = 1

2ln(2meγ

2v2Wmax

I 2) − β 2 −

δ (βγ )2

−C(I ,βγ )

Z

−dEdx

=4π k2e4

mec2

z2

β 2

ρZ NA

AB(v); B(v) = 1

2ln τ 2 (τ + 2)2(I / mec

2 )2+F(τ )2

−δ (β)2

−C(I ,β)Z

F(τ ) = 1− β 2 +

τ 2

8− (2τ +1)ln2

(τ +1)2, for electrons

F(τ ) = 2 ln2 − β 2

12[23+ 14

τ + 2+

10(τ + 2)2

+4

(τ + 2)3], for positrons

answer: 1/2 K.E.incoming and 1 K.E.incoming

Straggling for Electrons

• previous slide accounted for the possibility of large energy loss in a single collision and the effect of indistinguishability on the scattering cross section and kinematics in the modified Bethe-Bloch formula

• the straggling for electrons is observed to be “larger” than for heavy particles because each scattering event could have a significant fractional energy loss

• for thin absorbers, still can describe with Landau distribution

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Straggling Comparison: Electrons and Alphas

Radiative Energy Loss for Electrons

• Cherenkov radiation – small, but will look at because it is interesting

• bremsstrahlung – radiative loss becomes dominant at high energies

Cherenkov Radiation

• EM “shock wave” occurs when vparticle > c/n; with emission at cos θ = 1/βn

• the electric field of the particle interacts with atoms in the material and the particle is “right at” the atom before the field gets there

• Cherenkov spectrum goes as 1/λ2 (thus is peaked in the UV-blue)

• it’s the blue glow of a nuclear reactor

Aside: Radioactivity/Radiation, Does it Glow?

fiction

fact

Energy Loss from Cherenkov Radiation

• is already included in the Bethe-Bloch formula! how?

• remember the density effect? was the result of polarization and screening of distant atoms from the charged particle; more important at higher energies and more important for dense material

• as the charged particle zips through the material, polarization is induced and then relaxes to zero after the particle has gone by - results in an EM oscillation (or wave)

• the coherent sum at the shock wavefront, when the conditions are possible for Cherenkov emission, is part of the density effect calculation

• the dE/dx for the Cherenkov portion of the energy loss is only about 1% of the typical dE/dx value (in condensed materials)

Energy Loss from Bremsstrahlung

• occurs near the nucleus Coulomb field

• electron-electron bremsstrahlung is also possible

• emission probability (bremsstrahlung cross section):

• bremsstrahlung cross section dσ/dν can be calculated from first principles

σ ∝ ( e2

mc2)2

energy loss for e– and p in Cu

1 MeV

dσdν

= 4αZ 2 e2

mec2

⎛⎝⎜

⎞⎠⎟

21ν

screening plus…{ }

Φrad =1E0

hν dσdν(E0 ,ν)dν

0

ν0

∫

−dEdx

⎛⎝⎜

⎞⎠⎟ rad

= N E0Φrad

ν0 = E0 / h

but there is screening of the nuclear chargeby the surrounding atomic electrons

not significant for heavy particlesuntil very high energies

Bremsstrahlung Energy Loss Clarified

• bremsstrahlung cross section depends on mostly these terms

• with only a weak dependence on E0, the incident electron energy which is in the { } in the equation above• the fine structure constant α=1/137• ν is the frequency of the emitted bremsstrahlung photon

• the total energy loss by bremsstrahlung radiation would be the number of nuclear targets in the slice dx times the integral of the cross section times the energy per photon

• and since the cross section goes approximately as 1/ν, inside the integral there is no ν dependence and so integrates to terms times hν0 = E0

• thus, we can write

• which is just a way to say that dE/dX depends on E0 and an integral that doesn’t because it is written so that E0 cancels (approximately)

dσdν

= 4αZ 2 e2

mec2

⎛⎝⎜

⎞⎠⎟

21ν

screening plus…{ }

−dEdx

⎛⎝⎜

⎞⎠⎟ rad

= N hν dσdν(E0 ,ν)dν

0

ν0

∫

−dEdx

⎛⎝⎜

⎞⎠⎟ rad

= N E0Φrad ; Φrad =1E0

hν dσdν(E0 ,ν)dν

0

ν0

∫

Bremsstrahlung versus Ionization Energy Loss

• radiation loss increases linearly with E and goes as the square of Z

• ionization loss increases logarithmically with E and linearly with Z

• define critical energy:

• approximate formula for Ec (Bethe and Heitler):

• for E>Ec, radiation loss dominates and we re-write:

• defines the quantity radiation length Lrad:

dEdx

⎛⎝⎜

⎞⎠⎟ rad

=dEdx

⎛⎝⎜

⎞⎠⎟ collision

for E = Ec

Ec

1600mec2

Z

material Ec [MeV]Cu 24.8Pb 9.51air (STP) 102plastic 100water 92

−dEE

= NΦrad dx

E(x) = E0 exp−xLrad

⎛⎝⎜

⎞⎠⎟

material Lrad [cm]Cu 1.43Pb 0.56air (STP) 30050plastic 42.9water 36.1

Multiple Scattering

• multiple scattering causes the direction to change (and the total path length travelled to be much longer than the penetration depth) and can be significant for electrons

• normally dE/dx and scattering treated independently; but we know that’s not true as large angle scattering comes from large momentum (energy) transfer and a small angle scatter was a small energy transfer

• there are Gaussian approximation formulae describing angular distributions

no real point writing them out...

Backscattering

• because they are so light, the backscattering probability is significant for electrons and not so significant for heavy charged particles

• either from one large deflection or multiple scattered

• this is an important consideration for electron detectors; the backscattered electron deposits only a fraction of its energy in the detector

• backscatter fraction η depends on target thickness, reaching saturation at half the practical range

N U C L E A R I N S T R U M E N T S A N D M E T H O D S 94 ( I 9 7 I ) 5 o 9 - 5 t 3 ; '~: N O R T H - H O L L A N D P U B L I S H I N G CO.

AN EMPIRICAL EQUATION FOR THE BACKSCATTERING COEFFICIENT OF ELECTRONS

T. TABATA, R. ITO and S. OKABE

Radiation Center of Osaka Prefecture, Sakai, Osaka, Japan

Received 12 March 1971

Backscattering coefficient fl of monoenergctic electrons impinging normally on the thick target has been found to be expressed by an empirical equation of the form q = a~/(l +a2"r"3), where r is the incident kinetic energy in units of the rest energy of the electron, and the parameters a~ ( i = 1,2,3) are given by simple functions of target atomic number Z. Values of eight constants to

1. I n t r o d u c t i o n

In the detection and utilization of electrons, accurate knowledge of backscattering is frequently required. One of the quantities featuring this phenomenon is the backscattering coefficient defined as the ratio of the number of backscattered electrons to the number of incident electrons. The backscattering coefficients in- crease with increasing target thickness until saturation is reached at a thickness around half the practical range of the incident electrons. The backscattering coefficient r/ at saturation of the monoenergetic electrons im- pinging normally on the target is of general importance, and a good deal of experimental data for q have been reported i - 2 5),.

In order to obtain the value of r/ for the arbitrary atomic number Z of the target and the incident kinetic energy T of the electrons, it would be convenient to use the empirical equation which well reproduces the most probable values given by the existing data. In the

o . 3 ~ j f i i i i i T - - - - r ~ i I l l i r ! r T ~ - . 0.2 ~

- \ \

O.I

o . I I l o T (MeV) (a)

express ai have been determined by least-squares fit to a total of 615 experimental points reported in 20 references. This equation is valid for Z ~ > 6 in the energy region from about 50 keV to the highest energy of the existing experimental data (22 MeV), and the rms deviation of the experimental data from the equation is about 7 ~ .

energy region from about 10 to 100 keV, the coefficient fl is almost independent of T, and can be expressed as a function of Z only. Several equations to express the relation between q and Z have been formulated through simple theoretical treatments or semiempirical ap- proaches 26-3°) (a brief review of previous works is given in the appendix). At higher energies, the equa- tion for q should express the dependence on both Z and T. Although such equations have also been pro- posed,a, ~9,2~,3~), their regions of validity have rather inconvenient limitations. The present paper describes a new empirical equation valid in wider regions of Z and T than the previous ones.

2 . F o r m u l a t i o n

When experimental data for r/ are plotted as a function of T on a semilogarithmic paper, the data for T > 50 keV form S-shaped curves as shown in fig. 1. These curves suggest that the relation between ~1 and T may be expressed by an equation of the form

q = !al {1 - tgh [!(lna2 + a31nz)] }, * For energies below 50 keV, see the references listed in ref. 1.

0 . 6 ; 7 q - ~ - - ~ 7 i irqqqT~ r r ~ j l l l l l Au I 0.5 ~ - ~ ~ . ,. "

o 4 ! Agt " ~ ~

02 0:5 ""

0 , I - -

0 LLt_i l] O.I I ! I [ l l r l I I O T (MeV) (b)

Fig. I. Backscattering coefficient q of electrons as a function of incident energy 7". (a): C, AI, and Cu; (b): Ag and Au. Solid circle: experiment 2 ~1); cross: Monte Carlo calculationa:3); solid line: present empirical equation. The dashed lines in (b) indicate the lower limit T~ to the energy region considered in the least-squares fit. Experimental data around the curves of Ag and Au

at T > 15 MeV are for Cd and Pb, respectively.

509

N U C L E A R I N S T R U M E N T S A N D M E T H O D S 94 ( I 9 7 I ) 5 o 9 - 5 t 3 ; '~: N O R T H - H O L L A N D P U B L I S H I N G CO.

AN EMPIRICAL EQUATION FOR THE BACKSCATTERING COEFFICIENT OF ELECTRONS

T. TABATA, R. ITO and S. OKABE

Radiation Center of Osaka Prefecture, Sakai, Osaka, Japan

Received 12 March 1971

Backscattering coefficient fl of monoenergctic electrons impinging normally on the thick target has been found to be expressed by an empirical equation of the form q = a~/(l +a2"r"3), where r is the incident kinetic energy in units of the rest energy of the electron, and the parameters a~ ( i = 1,2,3) are given by simple functions of target atomic number Z. Values of eight constants to

1. I n t r o d u c t i o n

In the detection and utilization of electrons, accurate knowledge of backscattering is frequently required. One of the quantities featuring this phenomenon is the backscattering coefficient defined as the ratio of the number of backscattered electrons to the number of incident electrons. The backscattering coefficients in- crease with increasing target thickness until saturation is reached at a thickness around half the practical range of the incident electrons. The backscattering coefficient r/ at saturation of the monoenergetic electrons im- pinging normally on the target is of general importance, and a good deal of experimental data for q have been reported i - 2 5),.

In order to obtain the value of r/ for the arbitrary atomic number Z of the target and the incident kinetic energy T of the electrons, it would be convenient to use the empirical equation which well reproduces the most probable values given by the existing data. In the

o . 3 ~ j f i i i i i T - - - - r ~ i I l l i r ! r T ~ - . 0.2 ~

- \ \

O.I

o . I I l o T (MeV) (a)

express ai have been determined by least-squares fit to a total of 615 experimental points reported in 20 references. This equation is valid for Z ~ > 6 in the energy region from about 50 keV to the highest energy of the existing experimental data (22 MeV), and the rms deviation of the experimental data from the equation is about 7 ~ .

energy region from about 10 to 100 keV, the coefficient fl is almost independent of T, and can be expressed as a function of Z only. Several equations to express the relation between q and Z have been formulated through simple theoretical treatments or semiempirical ap- proaches 26-3°) (a brief review of previous works is given in the appendix). At higher energies, the equa- tion for q should express the dependence on both Z and T. Although such equations have also been pro- posed,a, ~9,2~,3~), their regions of validity have rather inconvenient limitations. The present paper describes a new empirical equation valid in wider regions of Z and T than the previous ones.

2 . F o r m u l a t i o n

When experimental data for r/ are plotted as a function of T on a semilogarithmic paper, the data for T > 50 keV form S-shaped curves as shown in fig. 1. These curves suggest that the relation between ~1 and T may be expressed by an equation of the form

q = !al {1 - tgh [!(lna2 + a31nz)] }, * For energies below 50 keV, see the references listed in ref. 1.

0 . 6 ; 7 q - ~ - - ~ 7 i irqqqT~ r r ~ j l l l l l Au I 0.5 ~ - ~ ~ . ,. "

o 4 ! Agt " ~ ~

02 0:5 ""

0 , I - -

0 LLt_i l] O.I I ! I [ l l r l I I O T (MeV) (b)

Fig. I. Backscattering coefficient q of electrons as a function of incident energy 7". (a): C, AI, and Cu; (b): Ag and Au. Solid circle: experiment 2 ~1); cross: Monte Carlo calculationa:3); solid line: present empirical equation. The dashed lines in (b) indicate the lower limit T~ to the energy region considered in the least-squares fit. Experimental data around the curves of Ag and Au

at T > 15 MeV are for Cd and Pb, respectively.

509