Download Pair (and Triplet) Production Effect:

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Pair (and Triplet) Production Effect:
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In both Pair and Triplet production, a positron (anti-electron) and an electron (or
“negatron”) are produced spontaneously as a photon interacts with a strong electric
field from either a nucleus (pair production) or an electron (triplet production).
Teinitial
+
hν
positron
Pair Production
nucleus
Teinitial
−
{ triplet Production
if electron instead
of nucleus }
Annihilation, if “in-flight” then
γs
T +final ≠ 0
e
θ+
θ−
final
Electron: often, Te−
=0
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These interactions are dominant at high incident photon energy: hν >> m0 c 2
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Nobel Prize in physics, 1948, went to P. Blackett’s bubble! It works by condensation
in a super-heated liquid (cloud chamber), or boiling in a super-cooled liquid (bubble
chamber). The particles create local trail of bubbles like airplanes make condensate
tracks.
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This effect also invoked to explain Hawking radiation.
Lecture 7 MP 501 Kissick 2016
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After some math that you should do, the threshold energy for these effects to
take place is:
(hν ) min
 2m0 c 2 

= 2m0 c 1 +
2 
2
Mc


2
Where, if:
M ≡ M nucleus >> m0 then it's pair production
M = m0
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then it's triplet production
Assuming that the recoil of the nucleus is small, the available kinetic energy is
simply
pair
Tavail = hν − (hν )min
Tavail = hν − 2m0 c 2 for pair and triplet production (2m0 c 2 = 1.022MeV )
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Focusing now on Pair Production:
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The mean kinetic energy given to each of the two particles is half of the available
kinetic energy [actually, the positron gets a bit more energy because of the push
from the positively charged nucleus].
Te ± =
Tavail
2
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The mean angle given to each of the two particles is with respect to the incident
photon direction is
m0c 2
θ± =
Te ±
with units:
[θ ± ] = radians
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The 1/T dependence is similar to bremsstrahlung ! {and that’s not all as we will see}.
Higher energy particles get more forward directed !
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Cross-section for Pair production:
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Bethe and Heitler (1934) derived the atomic differential cross-section as follows:
{per atom}
d aκ pair
dTe ±
z2
= σ0
P
Tavail
And all of the complications are in P.
-- The quantity, σ 0 , is defined as follows:
r02
σ0 ≡
= 4.80 x10 − 28 cm 2 / atom
137
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The quantity, r0, is the classical electron radius, and also represents the range of
the strong nuclear force:
e2
r0 =
= 2.8179 x10−15 m
2
m0c
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Note that 1/137 = α, the fine structure constant.
numbers in the universe !!
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See Appendix for this lecture for more information about the fine structure
constant.
One of the most important
Lecture 7 MP 501 Kissick 2016
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P is a function of photon energy and almost independent of atomic number, z, as
shown in the following figure from Attix, page 149:
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Notice the symmetry in the above figure: Energy not given to the positron is given
to the electron and the other way around …
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The cross-section is again obtained by integrating the differential cross-section:
a
κ pair = σ 0 z
Tavail
∫
2
P
T
T ± = 0 avail
d (Te ± ) = σ 0 z
With:
∫ d (T
e±
0
∫ Pd (T
e±
/ Tavail )
0
e
1
1
2
1
/ Tavail ) = 1 then, P = ∫ Pd (Te± / Tavail )
0
1
∫ d (T
e±
/ Tavail )
0
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Using the above to define an average P = P :
a
κ pair = σ 0 z 2 P
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And this P also has little z dependence.
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If the interaction is too far from the nucleus, then many orbital electrons will
screen the nuclear electric field.
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When screening can be neglected, there is no z dependence and just a weak
logarithmic dependence on energy:
P≈
28 2hν 218
ln
−
9 m0 c 2 27
(2m0 c 2 << hν << 137m0 c 2 z −1/3 )
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When screening is maximized, at high energy, there is a weak logarithmic z
dependence and basically no energy dependence:
28
2
ln(183z −1/3 ) −
9
27
(hν >> 137m0 c 2 z −1/3 )
P≈
(If z=6, this is hν>>35MeV)
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At energies around m0 c 2 , no analytical form is possible.
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We can use the approximation that
a
κ pair is proportional to z2 for all photon
energies.
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There is a great similarity between pair production and bremsstrahlung!!
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Aside: the similarity between pair production and bremsstrahlung:
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There is a concept about anti-particles, proposed by P. Dirac in 1930, that the
negative energy root from E 2 = ( pc) 2 + (m0c 2 ) 2 is an anti-particle. This came out of
the Schrödinger Equation with some relativity.
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A photon with enough energy, and an electric field to exchange momentum with, can
liberate something out of the infinite sea of negative energy (the “Dirac sea,”
completely filled and occupied states = vacuum!), and the hole left behind is the
anti-matter:
γ
energy
negatron
2m0c2
0
positron
“Dirac sea”
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Another mathematical oddity is that one can perhaps view the positron as moving
backwards in time. This is used in Feynman Diagrams where the similarity between
bremsstrahlung and pair production is profoundly obvious!
Pair production
time
Bremsstrahlung
time
electron
electron
positron
γ
γ
nucleus
γ
γ
position
nucleus
position
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The cross-sections (cm2/atom) are VERY similar:
a κ pair =
Bremsstrahlung (radiative losses):
r02 2  36
4
σγ =
z  ln(183 z −1/ 3 ) − 
137  9
18 
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σγ
a κ pair ≈
Therefore,
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The Pair Production Mass Attenuation coefficient is as follows then:
κ
N
=a κ A
A
ρ
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r02 2  28
2 
z  ln(183 z −1/ 3 ) − 
137  9
27 
Pair production at very high energies:

N 
cm 2
with units of  a κ ⋅ A  =
⋅
A  atom

(atoms / mole) cm 2
=
g / mole
g
Recall for Compton:
N z
N
σ
=e σ A ⇒a σ A
A
A
ρ
if a σ = z⋅e σ
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Focusing now on Triplet Production:
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The electric field is now from an electron, a very light particle which becomes
indistinguishable from the created particle!
Annihilation, if “in-flight” then
γs
T +final ≠ 0
Teinitial
+
hν
positron
e
θ+
θ−
Triplet Production
Teinitial
−
And another!
final
Electron: often, Te −
Teinitial
−
=0
Impossible to tell which is which !!!
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The mean kinetic energy given to each of the three particles is a third of the
available kinetic energy [actually, the positron still gets a bit more energy because
of the push from the positively charged nucleus that most available electrons are
finding themselves near].
Te −+− =
•
Tavail
3
 2m c 2 
Of course, since M = m0 in (hν ) min = 2m0c 2 1 + 0 2  , the threshold is now 4m0 c 2 !
 2Mc 
It is all due to momentum conservation.
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The Triplet Production Cross-section
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It would be the same but we actually use a factor, C, to relate the triplet crosssection to the pair production cross-section because of electron exchange effects:
a κ triplet =
Where,
a
κ pair
Cz
= σ0
zP
C
C = 1.6 ± 0.2 (5MeV < hν < 20MeV )
C = 1.1± 0.1 (20MeV < hν < 100MeV )
-- And C has a negligible dependence on z.
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Combined Triplet and Pair Cross-sections:
a
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κ = a κ pair + a κ triplet = σ 0 zP ( z + 1 / C )
Combined Triplet and Pair Mass Attenuation Coefficients:
κ  NA 
=
σ 0 zP ( z + 1 / C )
ρ  A 
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Combined Triplet and Pair Mass Energy Transfer Coefficients:
The fraction of photon energy transferred to charged particles is Tavail / hν ,
therefore,
κ tr κ Tavail κ  hν − 2m0c 2 

= ⋅
= 
ρ ρ hν
ρ
hν

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Not a typo! The 2m0c2 is for BOTH ! (Triplet has an extra momentum issue, but
still just creating 2 particles.) This is shared between either 2 or 3 particles.
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Right near the threshold, the amount of energy transferred is small, but this crosssection approaches the attenuation cross-section at large energies !
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Positron ‘annihilation in flight’:
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When the positron meets an electron, two gamma photons are released in opposite
directions in the center-of-mass (or ‘center-of-momentum’) frame. Therefore, an
isotropic angular distribution in this frame. Both photons have the same circular
polarization: both RHC or both LHC.
center-of-mass frame
laboratory frame
hν CofM
hν 1
e+
e-
e+
final
hν CofM
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e-
Te +
hν 2
The annihilation gammas in the lab frame will have a sum of energies:
hν 1 + hν 2 = Te +final + 2m0 c 2
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We will need to include the lost charged particle (positron) kinetic energy in the
calculation of µ en .
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The mass annihilation coefficient was derived by Heitler:
(
)
σ annihil N A z πr02  γ 2 + 4γ + 1 
γ +3 
2


γ
γ
=
⋅
ln
+
−
1
−


ρ
A (γ + 1)  γ 2 − 1 
γ 2 − 1 
Where: γ =
Te+final
m0c
2
+1 =
1
1 − (v / c) 2
{See Feynman, Chapter 18, ~0.1% of the time, 3 photons come off, not two, if
positronium had l (momentum) not equal to zero.}
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Note:
σ annihil (z / A)
∝ final
ρ
Te
+
for Te+final >> m0 c 2
So that annihilation is much more likely at low energies!
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“positronium” is name given to the temporary positron-electron thing that exists
for a short time when both have basically no kinetic energy – then they give off two
identical, opposite gammas of 0.511 gammas in either frame!
hν 1 = 0.511MeV
Te +final = 0
e+/-
hν 2 = 0.511MeV
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Next lecture: some extras like photonuclear reactions and Raleigh scattering.
Then, we start to form a cohesive picture of photon interactions.
Lecture 7 MP 501 Kissick 2016