What did you learn in the last lecture about accelerators?
What did you learn in the last lecture about accelerators?
Interlude – The concept of cross-section
For a thin target:
I
N = ( I )(σ )(t )
N= # of interactions per unit time
I = # of incident particles per unit time
σ = cross-section or interaction probability. It is usually expressed in cm2 (or barns).
t = target thickness expressed in # of target nuclei/cm2.
1 barn = 1 b = 1x10-24 cm2
Exercise
A target of 197Au 52 nm thick is bombarded by a 20 MeV proton beam with an intensity
of 1 x 108 p/s. The total reaction cross-section is 1.5 b. The density of Au is 19.3 g/cm3.
a) what is the number of interactions occurring in the target per second?
b) How does the total cross-section compare with the geometric cross-section of
the 197Au nucleus? Why can they be different?
INTERACTION OF RADIATION WITH MATTER
All radiation is detected through its interaction with matter!
INTRODUCTION:
What happens when radiation passes through matter?
Emphasis on what happens to emitted particle (if no nuclear
reaction and MEDIUM (i.e., atomic effects)
RELEVANCE:
(1)
(2)
(3)
(4)
(5)
Detection of Radiation
Radiation Safety
Environmental Hazards
Biological Effects − "Radiation Hypochondria"
Risk Assessment – Alternative Medicine
TYPES OF RADIATION:
(1)
(2)
(3)
(4)
Positive Ions:
Electrons:
Photons:
Neutrons:
I.
Positive Ions
Definition:
X+q − α, fission, cosmic rays, beams
β±, IC, Auger, cosmic rays
γ → x-ray → uv → visible
nuclear reactors, nuclear weapons, accelerators
Cation =
A +q
ZX
where q = atomic ionization state
Actual SRIM calculation of energy
loss as ions stop in matter.
A.
Overview
1.
Nuclei
Possible interactions:
σ ~ 10−24 cm2
Orbital e−s
σ nucleus
= 10 −8
σ electrons
Since
2.
σ ~ 10−16 cm2
Ion-electron collisions
dominate the interactions
Qualitative Properties of ion-electron Collisions
a.
b.
c.
vI < < c (usually ~ 0.01 − 0.1 c)
Mass (ion) > > Mass (e−) ; ∴ Many collisions required to stop ion
Trajectory: straight line
Analogy: bowling ball – ping-pong ball collisions
B.
Stages of Energy Loss
Electronic Stopping:
a.
Stripping:
A +q
ZX
ion
vI > > ve
→
→
A +Z
ZX
Medium
(electron sea)
in atomic orbitals 95% of c
; e.g.,
16 + 2
8O
→
→
16 + 8
8O
i.e., ion loses all electrons (usually) in passing through matter (∆X~100 atoms)
b.
Ion-Electron Collisions
Multiple, sequential collisions ; straight-line trajectory
c.
Medium Effects
(1) Ionization → Creation of multiple cations (from medium) – electron pairs
(2)
Electronic Excitation: fluorescence (uv, x-rays, etc.)
(3) Molecular Dissociation (free radical formation)
2.
Intermediate Stopping:
vI ≈ ve− (inner shells)
a.
Pickup: Incident ion begins to pick up electrons from stopping
medium. K-shell first, since they have highest velocity (binding energy).
b.
Moderate Directional Changes (Dramatic size increase)
O± 1,0
c.
Ion slows down at each step and ionic charge is ≈ neutralized
3.
Atomic ("nuclear") stopping: vI ≈ ve (valence shell)
a.
Ion charge ±1,0
b.
Elastic ion-atom collisions
∴ Mass (ion) ≈ Mass (medium atoms) – billiard ball collisions
Result large directional changes:
Straggling
c.
4.
Summary
Start
5.
Stop
Concept of Range (R represents Range and NOT Rate)
a. Definition:
b. Straggling:
The average distance traveled by an ion with a
given energy E during stopping process.
The distribution of ranges resulting from the statistical
nature of the stopping process
C.
1.
Energetics
Maximum energy loss per collision:
a.
∆Emax
∆Emax is obtained when ion scatters at 180° (c.m.)
From energy and momentum conservation (relativistic solution)
∆Emax = 4 E0 (Me/Mion)
b.
2.
=
E0/459 Aion (MeV)
Example: 6 MeV 4He ion
∆Emax = 6.000 /459(4) = 0.003 MeV
∴ E(α)′ = 6.000 −0.003
= 5.997 MeV; i.e., long way to go
Average Energy Loss:
<∆E>
Average over all scattering angles,
<∆E> ≈ 100 eV for 6.000 MeV α
< N collisions >=
E0
6.000
=
≈ 10 4 − 105
< ∆E > 0.0001
3.
Each collision creates a cation-electron pair; creates a measurable current; basis
for detectors
D.
Rate of Energy Loss: dE/dx: Specific Ionization (Related to radiation damage)
What can we observe by examining
these numbers?
1. Units:
dE MeV MeV 1 MeV cm3
MeV
=
=
• =
•
=
dx
cm
cm ρ
cm
g
mg / cm 2
Since ρ is constant
2. Bethe-Bloch Formula – For Positive Ions in Matter
Relativistically, by considering the momentum transfer to the electron (in the transverse
direction) one can derive (see FKMM or ES for derivation):
dE 4πZ 2e 4 n  2mv 2
2
2
β
β
ln
ln(
1
)
−
=
−
−
−


dx
mv 2 
I

v is the velocity of the ion
m is the mass of the ion
Z is the atomic number of the ion
v
β=
c
I is the ionization potential of the absorber
n is the number of electrons per unit volume in the absorber
This equation can be simplified in the non-relativistic case for
fully-stripped ions (γ = q/Z = 1) :
2
∆E γAZ ion
dE
∫ dx dx = ∆x ∝ Eion
Let’s do the following experiment
Z,A,E
We measure the following plot.
How is it related to what we have
learned ?
Si
CsI(Tl)
FUNDAMENTAL EQUATION OF RADIATION DAMAGE BY POSITIVE ION
a. Note: dE/dx increases with Z & A of ion
dE/dx decreases with E of ion
b.
Terminology:
dE/dx ≡ ionization ≡ energy loss ≡ radiation damage
4.
Result: Bragg Curve
Bragg peak – point at which maximum ionization occurs
BASIC PRINCIPLE OF RADIATION THERAPY
E.
Range Determination
Relation between Range and Specific Ionization:
0
dx
dE
R=∫
dE
E
1. Calculations:
Require knowledge of atomic orbital densities and
binding energies. Some success for light ions
2. Range Graphs for 1H and 4He
A 100 MeV p has a range
of 9000 mg/cm2. Density
of Al is 2700
mg/cm3.Therefore, the
thickness of Al required to
stop a 100 MeV proton is:
What thickness of Al is
necessary to stop a 500
MeV proton?
Why do the range curves for alpha particles and
protons diverge at low energy?
3.
Ranges of Other Ions in Al
a.
Scaling: Relative to protons
R(Zi, Ei, Ai) =
b.
A i Zp2
A p Zi 2
 A 
Rp(Ei/AI) =  2i  Rp (Ei/Ai)
 Zi 
Example: 500 MeV 20Ne ion
R(10,500 MeV ,20) =
20  500  1
1
mg
(
25
)
(
900
)
=
=
R
R
MeV


P
10 2  20  5
5
cm 2
R (500 MeV 20Ne) = 180 mg/cm2 , or 0.67 mm
4.
Methods exist to determine:
a.
b.
c.
dE/dx
Other absorbers
Compounds
we will not do this; procedures
similar to finding range
II. Electrons ( and Positrons prior to annihilation)
A.
B.
Sources
1. Radioactive Decay:
β±, IC, Auger, pair production
2.
Electron Accelerators:
Therapy, light sources
3.
Cosmic-Ray Showers:
lower atmosphere:
X+q
Energy-Loss Mechanism -- σ(nucleus)/σ(atom) ≈ 10−8 again
electron-electron collisions
billiard ball
1.
Ionization
a.
Repulsive charge-charge interaction
b.
Me = Me ; ∴ number of collisions much smaller
c.
Electrons relativistic above ~ 10 keV
d.
Products: scattered e− and cation-electron pair
NET RESULT: Greater energy loss per collision; ∴ greater straggling;
collisions less frequent ; ∴ ionization density much lower
C. Range Energy Relation
1.
Range determination – direct from graph
Notice that a 1 MeV β has a range of 400 mg/cm2 in Al. What energy alpha
particle has this same range?
2.
Absorber dependence
When the electron energy is low, the range is not a function of the absorber Z.
3.
Example:
10 MeV e− in Al
R e− ≈ 5500 mg/cm2 ≈ 2 cm
(Rα (10 MeV) ≈ 10 mg/cm2 = 0.004 cm)
The 10 mg/cm2 was estimated from range chart above.
D.
Bremsstrahlung
When the velocity of the electron is approximately c (ve ≈ c)
e− − e− interactions decrease ; long range.
∴ higher probability of passing in vicinity of a nucleus
∴ path is bent due to Coulomb interaction and energy is radiated
to conserve momentum
2.
Probability
EZ abs
P(bremsstrahlung )
=
800 MeV
P(ionization)
3.
4.
Result
a.
b.
High Z – good photon producer
Low Z – good shielding for high energy electrons.
Light Sources
Create same effect by passing an electron beam through a magnetic field H.
By adjusting H and Ee can fine tune Ephoton
Gives uv and x-ray sources of high intensity and variable frequency;
significant role in future chemical research
III.
Electromagnetic Radiation -- Photons
A.
Sources: Electromagnetic Spectrum
1.
Rearrangement of nuclear orbitals: γ-rays
2.
Rearrangement of atomic and molecular orbitals: x-rays, uv …
3.
Annihilation radiation ; e.g., e+ −e− → two 0.511 MeV γ s
4.
Bremsstrahlung: electron deceleration
5.
Cosmic ray showers
B.
Interactions
1. Photon:
Carriers of Electromagnetic force
∴ must interact with electric charge
Medium:
a.
b.
electrons
protons in nucleus
γ - e− most probable – size argument again
2.
Mechanisms
a.
Photoelectric Effect:
Eγ
b.
Compton Scattering:
Eγ
c.
photon disappears
photon scatters
Pair Production:
Eγ
e± pair produced
C.
Photoelectric Effect
1. Mechanism: Photon is completely absorbed by a charged particle; all
energy Eγ is transferred to an atomic electron, which is ejected from the atom.
2.
THEREFORE ONE COLLISION STOPS PHOTON
Ejected electron (photoelectron) is
monoenergetic.
3.
Ee = E γ − EB (n)
where EB(n) is
electron binding energy
for n orbital
When Eγ ≳ EB(n)
λγ ≈ λe-
∴ resonance-like situation; large wave function overlap leads to
high absorption → probability
4.
For Eγ ≳ EB
Z5
PPE = 7 / 2
Eγ ( MeV )
Best absorbers:
heavy elements (Pb)