Pair (And Triplet) Production Effect

Pair (and Triplet) Production Effect:

• 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).

Annihilation, if “in-flight” then γ final initial s T + ≠ 0 T + e e hν positron Pair Production θ+ θ nucleus − initial T − e final T − = 0 Electron: often, e { triplet Production if electron instead of nucleus }

• ν >> 2 These interactions are dominant at high incident photon energy: h m0c

• 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.

• This effect also invoked to explain Hawking radiation.

Lecture 7 MP 501 Kissick 2012

• After some math that you should do, the threshold energy for these effects to take place is:

 2m c2  ν = 2  + 0  (h )min 2m0c 1 2   2Mc 

Where, if:

≡ >> M M nucleus m0 then it' s pair production = M m0 then it' s triplet production

• Assuming that the recoil of the nucleus is small , the available kinetic energy is simply

= ν − ν pair Tavail h (h )min = ν − 2 2 = Tavail h 2m0c for pair and triplet production 2( m0c .1 022 MeV )

• Focusing now on Pair Production :

• 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].

Tavail T ± = e 2

Lecture 7 MP 501 Kissick 2012

• The mean angle given to each of the two particles is with respect to the incident photon direction is

2 m0c θ± = with units: [θ± ] = radians ± Te

• The 1/T dependence is similar to bremsstrahlung ! {and that’s not all as we will see}. Higher energy particles get more forward directed !

• Cross-section for Pair production :

• Bethe and Heitler (1934) derived the atomic differential cross-section as follows: {per atom} d κ z 2 a pair = σ P 0 ± dTe Tavail

And all of the complications are in P.

σ -- The quantity, 0 , is defined as follows: 2 r − σ ≡ 0 = 80.4 x10 28 cm 2 / atom 0 137

• The quantity, r 0, is the classical electron radius, and also represents the range of the strong nuclear force: 2 = e = −15 r0 2 56.2 x10 m m0c

• Note that 1/137 = α, the fine structure constant. One of the most important numbers in the universe !!

Lecture 7 MP 501 Kissick 2012

e 2 1 • A bit about the fine structure constant, α ≡ = : {Gaussian units} hc 137

• It was introduced by A. Sommerfeld in 1916.

• It measures the coupling strength between a photon and an electron.

r h • The Compton wavelength of an electron is λ = 0 = . e α 2 m0c r • The Bohr orbit of the hydrogen atom is 0 . α 2 • It also is related to the relative strength of the electromagnetic force:

Force: Relative strength: Notes:

Strong Nuclear 1 Nucleus/nucleons

Electromagnetic 1/137 Dominant at our scales

Weak Nuclear 10 -5 Explains β-decay*

Gravity 10 -39 The real oddball here !

*At > 100 GeV, this is inseparable from the electromagnetic force and becomes the ‘electroweak’ force. They can all hopefully combine this way, but gravity is the big challenge here eventually for a unified field theory.

• So, 1/137 is the relative strength of the electromagnetic force.

• And THEREFORE,

r 2 σ ≡ 0 expresses the area that can couple 0 137

electromag netic forces.

Lecture 7 MP 501 Kissick 2012

Back to the cross-section for pair production !

• P is a function of photon energy and almost independent of atomic number, z, as shown in the following figure from Attix, page 149:

• Notice the symmetry in the above figure: Energy not given to the positron is given to the electron and the other way around …

• The cross-section is again obtained by integrating the differential cross-section:

Tavail 1 2 P 2 κ = σ z d(T ± ) = σ z Pd (T ± /T ) a pair 0 ∫ T e 0 ∫ e avail T ± =0 avail 0 e

1 1 1

d(T ± /T ) =1 P = Pd (T ± /T ) d(T ± /T ) With: ∫ e avail then, ∫ e avail ∫ e avail 0 0 0

Lecture 7 MP 501 Kissick 2012

• Using the above to define an average P = P :

κ = σ 2 a pair 0 z P

• And this P also has little z dependence .

• If the interaction is too far from the nucleus, then many orbital electrons will screen the nuclear electric field.

• When screening can be neglected, there is no z dependence and just a weak logarithmic dependence on energy:

28 2hν 218 P ≈ ln − 9 m c2 27 0 2 << ν << 2 − 3/1 2( m0c h 137 m0c z )

• When screening is maximized, at high energy, there is a weak logarithmic z dependence and basically no energy dependence:

28 − 2 P ≈ ln( 183 z 3/1 ) − 9 27 ν >> 2 − 3/1 (h 137 m0c z ) (If z=6, this is h ν>>35MeV)

• 2 At energies around m0c , no analytical form is possible.

• κ 2 We can use the approximation that a pair is proportional to z for all photon energies.

• There is a great similarity between pair production and bremsstrahlung !!

Lecture 7 MP 501 Kissick 2012

• Aside: the similarity between pair production and bremsstrahlung:

• There is a concept about anti-particles, proposed by P. Dirac in 1930, that the 2 = 2 + 2 2 negative energy root from E ( pc ) (m0c ) is an anti-particle. This came out of the Schoedinger Equation with some relativity.

• 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: energy γ negatron

2 0 2m 0c

positron

“Dirac sea”

• 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 Bremsstrahlung time time

electron electron positron

γ γ γ nucleus γ nucleus

position position

Lecture 7 MP 501 Kissick 2012

• The cross-sections (cm 2/atom) are VERY similar:

2 r 2  28 − 3/1 2  Pair production at very high energies: κ = 0 z  ln( 183 z ) −  a pair 137  9 27 

2   σ = r0 2 36 − 3/1 − 4 Bremsstrahlung (radiative losses): γ z  ln( 183 z )  137  9 18 

7 κ ≈ σ γ Therefore, a pair 9

• The Pair Production Mass Attenuation coefficient is as follows then:

κ N A 2 2 = κ  N A  cm (atoms / mole ) cm a with units of κ ⋅ = ⋅ = ρ A  a A  atom g / mole g

{no z dependence here.}

• Recall for Compton:

σ = σ N A z σ N A e ⇒a σ = z⋅ σ ρ A A if a e

Lecture 7 MP 501 Kissick 2012

• Focusing now on Triplet Production :

• The electric field is now from an electron, a very light particle which becomes indistinguishable from the created particle !

Annihilation, if “in-flight” then γ final initial s T + ≠ 0 T + e e hν positron Triplet Production θ+ initial θ T − e − And another! final = initial Electron: often, T − 0 T − e e Impossible to tell which is which !!!

• 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].

Tavail T −+− = e 3

2  2m c  2 • M = m ν = 2  + 0  , 4m c ! Of course, since 0 in (h )min 2m0c 1 2  the threshold is now 0  2Mc  It is all due to momentum conservation.

2 Aside: Because of atomic excitations, the threshold is actually 2m0c , but very very small…

Lecture 7 MP 501 Kissick 2012

• The Triplet production Cross-section

• It would be the same but we actually use a factor, C, to relate the triplet cross- section to the pair production cross-section because of electron exchange effects:

κ κ = a pair = σ zP a triplet Cz 0 C Where, C = 6.1 ± 2.0 5( MeV < hν < 20 MeV )

C = 1.1 ± 1.0 20( MeV < hν <100 MeV ) -- And C has a negligible dependence on z.

• Combined Triplet and pair Attenuation Coefficients:

κ = κ + κ = σ + a a pair a triplet 0 zP(z /1 C)

• Combined Triplet and pair Mass Attenuation Coefficients:

κ  N  =  A σ zP(z + /1 C) ρ  A  0

• Combined Triplet and pair Mass Energy Transfer Cross-sections:

ν The fraction of photon energy transferred to charged particles is Tavail / h , therefore,

κ κ T κ  hν − 2m c2  tr = ⋅ avail =  0    ρ ρ hν ρ  hν 

• 2 Not a typo! The 2m 0c is for BOTH ! (Triplet has an extra momentum issue, but still just creating 2 particles.)

• Right near the threshold, the amount of energy transferred is small, but this cross- section approaches the attenuation cross-section at large energies !

Lecture 7 MP 501 Kissick 2012

• Positron ‘annihilation in flight’:

• 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

ν h CofM ν h 1

e+ e- e+ e- final T + e ν h CofM ν h 2

• The annihilation gammas in the lab frame will have a sum of energies:

final 2 hν + hν = T + + 2m c 1 2 e 0

• We will need to include the lost charged particle (positron) kinetic energy in the µ calculation of en .

• The mass annihilation coefficient was derived by Heitler:

σ N z πr 2  γ 2 + 4γ +1 γ + 3  annihil = A ⋅ 0  ln ()γ + γ 2 −1 −  ρ γ +  γ 2 −  2 A ( )1  1  γ −1

final T + 1 Where: γ = e +1 = 2 2 m0c 1− (v / c)

{See Feynman, Chapter 18, ~0.1% of the time, 3 photons come off, not two, if positronium had l (momentum) not equal to zero.}

Lecture 7 MP 501 Kissick 2012

σ annihil (z / A) final 2 ∝ for T + >> m c • Note: final e 0 ρ T + e

So that annihilation is much more likely at low energies!

• “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!

ν = h 1 .0 511 MeV

final e+/- T + = 0 e

ν = h 2 .0 511 MeV

• 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 2012