Cavity Theory and Interface Effects
Eirik Malinen
Definition of absorbed dose
D is the expectation value of the energy imparted to matter per unit mass at a point
• Is this an unambiguous definition?
• Two different media in the same radiation field will not receive the same dose
D is the expectation value of the energy imparted to matter per unit mass at a point in a given medium
dm D = dε
Cavity
• Cavity theory, narrow sense: convert “dose to detector” to “dose to medium”
• Cavity theory, broad sense: dose distribution in inhomogeneous media
med cav
γ, n, e-, …
Absorbed dose in γ irradiated thin foil, CPE
Energy transferred:
∆x
γ
e− e−
( )ν
γ
h N Rin
=
, Rin,e Rout,e Rout,γ
Rin,γ: Radiant incoming energy
= N(hν) for monoenergetic photons
x h
N
R R
R R
R R
tr
out in
CPE e out out
e in in
tr
∆
=
−
=
−
− +
=
µ ν
ε γ γ γ γ
) (
, ,
, ,
, ,
1 e−µ∆x ≈
Absorbed dose in γ irradiated thin foil, CPE
Absorbed dose (no brehmsstrahlung)
If brehmsstrahlung:
Ψ
∆ =
∆
= Ψ
= ∆
=
= ρ
µ ρ
µ µ
ν
εtr tr tr tr
x A
x A m
x h
N K m
D ( )
=
=Kc Ψ µρen D
µ ν
µ h
T
tr =
Energy loss from electrons
• Stopping power:
• Collision stopping power: Scol
• Restricted stopping power: L∆
∫
= +
=
= max
min
E
E
tot rad
col dE
dE E d n S
dx S
S dT ρ σ
∫
= max
min
E
E
col
col dE
dE E d n
S ρ σ
∫
∆∆
= d col dE E
n
L ρ σ
n: number of electrons per gram
Absorbed dose in thin foil, electrons
Energy loss <∆T> → energy imparted ε ?
→ Brehmsstrahlung, δrays, path lengthening Brehmsstrahlung: Srad
ℜ
<<
∆x
∆x Range of charged particle
Path lengthening due to multiple scattering
Scattering power:
∆
= dx
xd cos
) cos(
θ2
ρ θ
∆x θ
θ cos ' x x = ∆
∆ dx
dθ2
δ rays
• Energetic, secondary electrons
• Significant range compared to foil thickness
• Results from high energy transfers (included in Scol)
Maximum energy transfer:
2 2 2
max 2 1
β β
= m c −
E e
Heavy ions
2 / T Emax =
Electrons
δ rays
Energy imparted for charged particles:
δparticle equilibrium
δPE requirements: homogeneous medium and δPE always present under CPE
δ
ε =Rin,p+Rin,δ −Rout,p−Rout,
p , out p , in ,
out ,
in R R R
R δ = δ ⇒ ε = − ∆x
ℜp
<<
ℜδ
primary
δ rays
• Since βis low for heavy charged particles in the MeV- region, Emax is low
• β=0.1 (e.g. 38 MeV α-particles) gives Emax=10 keV
• Range of 10 keV electrons in water: 2.5 µm
• →δ-electrons deposit their energy locally, and δ- equilibrium may often be present
• Range of 1 MeV electrons: 0.5 cm
• →δ-equilibrium may not obtained for high energy electron beam
Absorbed dose
• Under δPE (foil sandwiched, short ), no path lengthening, no brehmstrahlung:
D= mε
A , N
S A N x A
x NS V
x D NS
x NS R
R
Rin,p out,p p
=
=
=
=
⇒
=
=
−
=
ρ Φ
∆ ρ
∆ ρ
∆
∆
∆ ε
ℜδ
= Φ ρS
D Fluence of primary electrons
Absorbed dose, thick foil, heavy particles
• The average dose may be found by:
– Calculating the residual range: ℜres= ℜin-L – Find the energy Toutcorresponding to ℜres – Imparted energy is: ∆T = Tin-Tout
– Dose:
L
L
Tin Tout
N T T
D m L
∆ ∆
= = Φ ρ
Foil placed in vacuum
δrays with T > ∆lost from foil (δPE absent):
=
−
−
=R R R N n
∫
EddEdEEmin
, out p , out p , in
∆
δ ρ σ
ε
= Φ Lρ∆ D
L
∆air water
∆=10 keV
Range and projected range, electrons
Spectrum of charged particles, δ PE present
ΦTdT : number of primary electrons cm-2in [ T, T+dT ] Minimum energy: 0
Maximum energy: Tmax
∫
=
⇒
=
⇒
Tmax
0 T T
dT S S D
dT
dD Φ ρ
Φ ρ
∫
= max
T
0
T S dT
D Φ ρ
Partial δ PE
Electron beams: constant fluence of secondary, low energy electrons with T < ∆
Energetic secondary electrons added to total fluence:
Particles either assigned to radiation field or to
∫
= max +
T p
T L dT
D
∆ δ ∆
Φ ρ
?
p
T
Φ +δ
Bragg-Gray cavity theory
B-G conditions:
1. Charged particle fluence is not perturbed by cavity 2. Absorbed dose entirely due to charged particles
wall cav
e-,p, ....
wall wall
cav cav
S D
S D
=
=
Φ ρ Φ ρ
cav
wall wall
cav S
D
D
=
⇒ ρ
Bragg-Gray cavity theory
Bragg-Gray-Laurence
Laurence: incorporated slowing down spectrum of charged particles generated in the wall
wall 0 T
T
0
0 wall T
T
0 0 T
0 wall
T
0 0 T CPE
0 wall
T
S n
0 dT S n
dT n S dT
T n S dT
D
0
0 0
0
ρ
= Φ
= ⇒
−
Φ ρ
⇒
=
Φ ρ
⇒
=
Φ ρ
=
∫
∫
∫
∫
Photons give rise tomonoenergetic electrons with kinetic energy T0
Bragg-Gray-Laurence
Slowing down spectrum of primary electrons in water
Bragg-Gray-Laurence
The total fluence:
Dose to cavity:
CSDA 0 T
0 0 T
0
T n
) / S ( n dT dT
0
0 = ℜ
= ρ Φ
=
Φ
∫ ∫
∫
∫
∫
=
=
= 0 0 0
T
0
cav
wall 0
T
0
wall cav 0
T
0 cav
T
cav S dT
n S dT
S n S dT
D ρ
ρ ρ Φ ρ
Burlin cavity theory
Absorbed dose
wall cavity
cav en
ρ Ψ µ
wall en
ρ Ψ µ
Burlin cavity theory
cav
wall wall cav cav
wall Small: Bragg-Gray Intermediate: Burlin Large: CPE
Burlin cavity theory
Cavity with dimensions << electron range: B-G theory:
Cavity with dimensions >> electron range: CPE-theory:
cav
wall wall
cav S
D
D
≈ ρ
cav
wall en
wall cav
D
D
= ρ µ
Burlin cavity theory
General theory for intermediate sized cavities:
d: average attenuation of electrons generated in the wall crossing the cavity
cav
wall en cav
wall wall
cav S d
D d
D
−
+
=
ρ µ ρ (1 )
L e d L
L e dx
dx e d
L L
L L
x
β β β
β β
β
1 1 1
0
0 = − ⇒ − = + −
= − −
−
∫
∫
Burlin cavity theory
β: effective electron attenuation coefficient Empirical expression:
tmax: depth at which 1 % of electrons can travel tmax/ℜCSDA ≈0.9 low Z
tmax/ ℜCSDA≈0.8 intermediate Z tmax/ ℜCSDA≈0.7 high Z
04 0.
e−βtmax ≈
Burlin cavity theory - assumptions
• Wall and cavity homogenous
• No significant γattenuation
• CPE exists
• Spectrum of δ rays equal in wall and cavity
• Electrons generated in wall are exponentially attenuated within cavity
• Electrons generated in cavity increase exponentially
Burlin cavity theory – experiment vs theory
60Co γrays
Perspex dosimeter in tin wall Corrected for attenuation Normalised to 8 cm
Interface dosimetry
Interface dosimetry
1 MeV γ
MC sim ΦAl
ΦC
Fluence considerations
Total equilibrium fluence, secondary electrons, CPE:
n0: number of electrons generated per gram
→
nℜCSDA
= Φ 0
CSDA 0
en tr
en CPE 0
n
h , h
h T , T
n D
ρℜ
∝µ Φ
⇒
ρ
∝µ
⇒
ν
∝ µ Ψ
νµ µ ≈ νµ ρ =
Ψµ
=
=
Fluence considerations
Therefore, fluence ratio, medium 1 and 2 becomes:
1 MeV γrays:
= 0.45 MeV , = 0.064 cm-1, = 0.061 cm-1
= 0.186 g/cm2 , = 0.211 MeV cm2/g
Φ /Φ ≈
( )
12 12 2 1
ℜCSDA
= Φ Φ
ρ µ
T
C
ρ µ
Al
ρ µ
ℜC ℜAl
Interface dosimetry
At the interface, transition from Φ1to Φ2 Simplistic vector representation:
Forward/backward ratio depend on medium
forward backward
ΦF ΦB
Backscatter ratio
Interface dosimetry
Simplistic considerations – total fluence ΦB+ΦF
At interface, Φtot≈ΦB,2+ ΦF,1
ΦF,1
ΦB, 1 ΦB, 2 ΦF,2
Φ1
Φ2
ℜ1 ℜ2
Depth (cm)
Interface dosimetry
1 MeV γ
MC sim ΦAl
ΦC
Interface dosimetry – change order of media
1 MeV γ
MC sim
