1
Charged Particle Interactions with Matter, Con’t:
•
Hard Collision Mass Stopping Power:
•
A hard collision is when an electron is ejected with a considerable fraction of the
maximum energy transferrable, T’max.
•
These recoil electrons are called “delta-rays” or “knock-on” electrons and they can
be seen in bubble chambers. Here is a view from Anderson, page 28:
•
The “short track” shown above is a hard collision.
•
There is some dependence on particle spin for this differential hard collision
cross-section per electron. We will use the form for zero spin particles (like
α, π,…), but it will apply to spin ½ particles (like protons, electrons, µ,…) provided
T’<<Mc2. Also, technically, this is for heavy charged particles.
2
2
dσ h
2
2 Z  1 − β (T ' / T ' max ) 

= 2πr0 mc 2 
dT '
(T ' ) 2
β 

Lecture 10 MP 501 Kissick 2016
2
•
Then get the hard collision mass stopping power:
T'

 dTh   N A z  max dσ h

 = 
T ' dT '
 ∫
 ρdx  c  A  H dT '

T 'max
T 'max

2
= k  ∫ dT ' / T ' − ( β / T 'max ) ∫ dT '
H
 H

[
= k [ln(T '
= k ln(T 'max / H ) − ( β 2 / T 'max )(T 'max − H )
•
max
/ H)− β
2
] for H << T '
max
Now we can combine things -- mass stopping power for heavy charged particles:
 dT 

 =
 ρdx  c
 dTs 


 ρdx  c
 dTh 


 ρdx  c
+
  2mc 2 β 2


= k ln 2
H  − β 2  + k ln(T 'max / H ) − β 2
2
  I (1 − β ) 

[
•
]
With a bit of math [ln(1/A)=-lnA, and ln(AB)=lnA+lnB], and recall for a heavy
charged particle (page 6 of last lecture),
•
]
T 'max ≈ 2mc 2
β2
≈ 2mv 2
2
1− β
2
the above simplifies: recall, important stuff in k ≡ (0.1535) z ⋅ Z ( MeV /( g / cm 2 ))
 A β2 


  2mc 2 β 2 

 dT 
2


 = 2k ln
−
β

2 
 ρdx  c
  I (1 − β ) 

•
A more handy form: units = MeV/(g/cm2) with I in eV ! {needs Born approx
valid and OK for spin ½ if T’ << Mc2 }
2

 β2 
 dT 
 z   Z 
2


 = (0.3071)    2  13.84 + ln
−
β
−
ln
I

A   β 
1 − β 2 
{
 ρdx  c


123
medium particle
Lecture 10 MP 501 Kissick 2016
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Shell Correction (when Born approximation is not valid):
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The complex motion of the orbital electrons is accounted for with this factor.
•
When the particle velocity is less than the orbital velocity of the electrons in that
shell, then those electrons do not participate in collisions with the particle.
Where Born approx. on edge of validity …
KE ~ PE :
Recall:
⇒M
v2
e 2 Me 2
~ zZ
= 2 zZe 2
2
a0
h
⇒ β 2 ~ ( zZ )
•
e4
zZ
=
2 2
h c
(137) 2
The shell correction factor is written as C/z, and it (mostly) decreases the mass
collisional stopping power by a small amount: (recall the dependences in k)
  2mc 2 β 2 

 dT 
2


 = 2k ln
−
β
−
C
/
z

2 
ρ
β
dx
I
(
1
−
)

c

 

•
The shell correction is a function of the particle velocity and the atomic number of
the medium, z (but rather complex function of z). It has a magnitude as follows
from this picture from Anderson:
Lecture 10 MP 501 Kissick 2016
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Radiative Stopping Power for Heavy Charged Particles:
•
A non-topic in the sense that heavy charged particles do not produce much
bremsstrahlung: (i.e., a proton is ~ 2000 times heavier than an electron, and
therefore 4x106 times less bremsstrahlung is produced.
 dT 
1
≈ 0 if M is much heavier than an electron

 ~
2 2
 ρdx  r ( Mc )
•
Dependence of the Stopping Power on the Medium:
•
The factor z/A has a value near 0.5 +/- 0.05 for most elements, dropping to
lower values at higher z. Hydrogen-1 has the highest value of 1 which is why it is
used to slow or shield fast charged particles. Remember this !! See below (Rock’s
handwriting here):
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The lnI term also makes high z materials have lower stopping power. Also, the shell
correction also decreases the stopping power generally.
Lecture 10 MP 501 Kissick 2016
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Mass Collisional Stopping Power Dependence on Particle Velocity:
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The 1 / β 2 is dominant at low energies! Remember this !!
•
It causes the mass collisional stopping power to sharply increase as the particle
slows down (see fig. 8.2 in Attix, page 169). The result is the so-called: “Bragg
Peak:” (picture from Radiobiology for the Radiologist, E. Hall)
•
Recall how to calculate between energy (kinetic) and speed (β):
(γ − 1) = T / Mc 2 & γ = 1 / 1 − β 2 & β ≡ v/c
•
When β ≈ 1 , the energy increases quickly as the speed creeps up against c. The
linac accelerating microwave cavity uses this fact to keep the particle in phase with
the phase velocity of the standing or travelling microwave. The cavity is loaded
with periodic barriers to slow the phase velocity below c.
Lecture 10 MP 501 Kissick 2016
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Also when β ≈ 1 , the 1 / β 2 term has little influence, but the β 2 /(1 − β 2 ) term
increases: i.e., for protons with kinetic energy Tp:
β
•
β2
Tp
(
ln β 2 /(1 − β 2 )
)
(
)
ln β 2 /(1 − β 2 ) − β 2
0.80
626
0.64
0.57
-0.07
0.90
1214
0.810
1.45
1.17
0.95
2067
0.903
2.23
1.33
0.99
5713
0.980
3.89
2.91
The kinetic energy of two particles with the same relative velocity, β , is directly
proportional to their rest masses.
•
There is no mass dependence on the heavy charged particle stopping power!
Therefore, any heavy particle with the same velocity and charge will have the same
stopping power, BUT THE SCATTER AND RANGE STRAGGLING COULD BE
DIFFERENT !
•
Mass Collisional Stopping Power Dependence on particle Charge:
•
Note that there is an effective charge, Z*, to be used instead of the charge, Z, at
low energies (speeds β < 0.1 ) because of the attachment of the incident particle’s
electrons. Higher energy particles are more likely to be fully ionized: (from
Anderson, page 21):
Lecture 10 MP 501 Kissick 2016
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The Z2 factor means that multiply-charged particles have a much higher mass
collisional stopping power than singly charged particles. For example:
If βα 2+ = β p + , then:
•
 dT  α 2+
 dT  p+


= 4 ⋅ 

ρ
dx
ρ
dx

c

c
This fact can be used to obtain stopping powers for any heavy charged particle
from a table of mass collisional stopping powers for protons. Do:
1. Look up or calculate β for particle x with kinetic energy, Tx.
2. Look up or calculate the proton kinetic energy, Tp, for the same β.
3. Look up the mass stopping power for a proton with kinetic energy, Tp.
4. Multiply the mass stopping power for a proton by ( Z x* / Z *p ) 2 . Note that Z *p is
unity, and Z x* is the effective charge on particle x at its speed.
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Next lectures: MORE on stopping power, especially for its tabulation and use for
our purposes.
Lecture 10 MP 501 Kissick 2016