In the previous blog post we mentioned that Frahm et al. (2009) have measured pair production when a photon hits a gadolinium nucleus. The energy of the photon in their measurements is moderately over the threshold energy 1.022 MeV.
We asked why the cross section at 1.1 MeV is only 1 / 100 of the cross section at 2 MeV incident photon energy.
Pair production is the dominant process when > 10 MeV photons hit atoms.
gamma ray photon
_______ e-
/
~~~~~~~
\________ e+
|
| virtual photon
Z+ --------------------------
nucleus
The photon gives up energy and some momentum to create the real electron. The photon transfers the rest of its momentum and energy to the created positron which then has too much momentum to be on-shell.
The positron bounces from the nucleus and the excess momentum is absorbed by the nucleus. The positron becomes on-shell, that is, real.
Our various classical models claim that the potential energy of the combined field of the photon and the nucleus can be reduced by creating a pair. Electric field lines "break" and the loose ends are the particles being created.
Our classical models do not make any numerical prediction for the cross section. Intuitively, creating a pair from a photon whose energy is only a little over 1.022 MeV should be much harder than from a 2 MeV photon. The result of Frahm et al. makes sense from the classical point of view.
We remarked in a blog post about classical virtual particles that the momentum and energy of the pair can in interaction with the nucleus be equal to the momentum and energy of a single photon. In principle, we could calculate the probability of pair production in this classical model, if we assume that the photon particle has a certain probability to split into a pair under a time interval.
A wave description of the Feynman diagram
The Feynman diagram can be understood with the following plane wave model:
t
^
| ---------- on-shell
| ---------- positron wave e+
| ● Z+
| | | | Fourier component
| | | | of static Coulomb
| | | | field of nucleus
|
| | | | off-shell
| | | | electron wave e+-
| / --- / --- / ---------
| / --- / --- / ---------- on-shell
| / --- / --- / ----------- electron
| / --- / --- / ------------ wave e-
| gamma laser
| beam wave
------------------------------------> x
The nucleus Z+ is not really located anywhere in the diagram. We work with the Fourier components of its Coulomb field. The components are plane waves which span the entire diagram at a constant amplitude, regardless of the position of the nucleus.
1. We assume a coherent laser beam of gamma photons. The oblique lines are the crests of the electromagnetic waves.
2. We assume a coherent flux of electrons with little momentum. The electron wave crest lines are almost horizontal because the momentum is small.
3. The strange thing in this model is that the electrons arrive from the future, or are assumed "just to exist there." Alternatively, we may assume that the electron wave is created "when needed".
4. The interaction term between the electromagnetic field and the Dirac field makes a source to the Dirac field. The source produces off-shell (virtual) electron waves. Another way to say this is that electrons absorb photons from the laser beam. The wave crest lines are almost vertical, because the off-shell electron has too much momentum relative to its energy to be real.
5. The interaction term between the off-shell electron wave and a Fourier component of the Coulomb field of the nucleus Z+ makes another source to the Dirac field. That source can produce on-shell positrons. We may say that the off-shell electron absorbed a virtual photon from the Coulomb field of the nucleus and became an on-shell positron.
For clarity, we did not superimpose all waves on each other in the diagram above. They should be superimposed in the accurate diagram.
We may imagine that the input plane waves are standardized, e.g., one particle per square meter per second.
How far can the virtual electron fly?
The diagram above is a momentum space description of the process. Points (t, x) are not genuine spacetime locations but just "substance" on which we can visualize plane waves. Particles in momentum space must be described as plane waves.
What would a position space diagram look like? We can draw the nucleus Z+ as a classical particle because it is very heavy.
The positron in the diagram must come very close to the nucleus to lose the extra momentum. If the nucleus is gadolinium (Z = 64), then the minimum distance should be around 64 * 1.4 * 10^-15 m = 10^-13 m, which corresponds to potential energy 1 MeV for the positron and the same negative potential for the electron.
A. The Compton wavelength of a real electron is 2 * 10^-12 m, which is quite a long distance. In an earlier blog posting we calculated from the time-energy uncertainty relation that the "lifelength" of a virtual particle might be 0.1 Compton wavelengths. That would be 2 * 10^-13 m.
B. Our classical model of virtual particles says that they must exist quite deep in the Coulomb field. That suggests that the virtual electron is ~ 10^-13 m from the nucleus.
C. The pair is born from some kind of a source in the Dirac equation. The source presumably is a result from some kind of interaction between the Coulomb field of the nucleus and the photon. That would suggest that the pair is born "close" to the nucleus.
D. We had a classical model where electric field lines break and the loose ends are the created particles. That suggests a "close" birh to the nucleus.
E. The inverse reaction is pair annihilation close to the nucleus, so that only one photon is produced, instead of two. Our classical model of annihilation suggests that it happens in a volume of size ~ 1.4 * 10^-15 m. This is evidence for a short lifelength for the virtual electron.
F. If we turn and twist the diagram, the positron seems to absorb a virtual photon from the nucleus, and emit a real photon. Our classical model of Thomson/Compton scattering suggests that the absorption and the emission are simultaneous. This is evidence for a short lifelength of the virtual electron.
G. Bethe and Heitler in their 1934 paper, in the case of bremsstrahlung (page 84), write that the transition occurs under the simultaneous action of the perturbations by the nucleus Coulomb field and the photon field of the produced photon. This suggests that the lifelength of the virtual electron is short.
F. If we try to measure experimentally where the electron and the positron are born, we have to face an uncertainty relation about the position and momentum. We may observe the created positron annihilating with the orbital electron of the atom around the nucleus. That way we might be able to localize the positron with an accuracy 10^-10 m. This, of course, requires that we assume that an orbital electron is a particle and has a definite location.
Conclusions: our classical models suggest that the virtual electron only flies 10^-15 m ... 10^-13 m. The uncertainty principle model suggests a lifelength up to 2 * 10^-13 m. Experimentally, we can hope to restrict the lifelength to < 10^-10 m.
In the plane wave diagram, the virtual electron seems to fly an arbitrarily long "distance", but that is just an illusion of the momentum space model.
Cross section for bremsstrahlung versus pair production from a photon
Feynman integral formulas must take into account the small volume of the actual position space diagram of the process. The virtual electron probably flies only a short distance.
The probability amplitude of pair creation is curtailed by the propagators of the virtual electron and the virtual photon, as well as the small coupling constant ~ sqrt(α) at the vertices.
The cross section for an on-shell positron flying within 10^-13 m from a gadolinium nucleus (potential > 1 MeV) is 300 barn.
Frahm et al. measured that the cross section for pair production with a 2 MeV photon is 0.2 barn.
Thus, the cross section for pair production is just 1 / 1,500 of the cross section of a "close hit".
If we turn and twist the Feynman diagram, we get one for bremsstrahlung of a relativistic electron:
~~~~~ 1 MeV photon
/
1.7 MeV e- ------------------------------ 0.7 MeV e-
| virtual
| photon
Z+ --------------------------------------
Let us check what is a typical cross section in bremsstrahlung in the MeV range.
S. H. Morgan, Jr. (1970) in Figure 9 has
dσ / dE = 10 barn / MeV
for Z = 50 and incident electron kinetic energy 1.7 MeV, where we require that the photon energy is around 1 MeV.
The cross section for the electron coming to a potential < -1 MeV is 200 barn.
The cross section for a 1 MeV photon production is 1 / 20 of the cross section of a close hit.
What is the cross section for inverse reactions?
A classical calculation as well as a Feynman diagram calculation tell us that the cross section for a particle to make a "close" hit (1 MeV Coulomb potential) to a typical nucleus is ~ 250 barn.
Empirical data as well as calculations by Bethe and Heitler (1934) show that when a 2 MeV photon hits a typical nucleus, the cross section for pair production is ~ 0.2 barn.
For bremsstrahlung producing a ~ 1 MeV photon, the cross section is larger, ~ 10 barn.
Why is a pair harder to produce? In pair production, "kinetic" energy of the photon is converted to "potential" energy of the pair: their masses. Physical systems generally tend to increase kinetic energy and reduce potential energy.
What about the inverse reactions? An inverse reaction requires three particles to meet. Or, at least, first 2 particles have to meet and after that, 2 particles have to meet again. The probability for an inverse reaction is extremely small unless the density of particles is huge.
We may say that the entropy of a system grows in a reaction if the inverse reaction is less probable. A random soup of particles tries to increase its entropy.
--------------------
| |
-----| |
| • hole |
-----| |
---------------------
small large vessel
vessel
Let us think of a small vessel which is connected to a large vessel through a narrow hole. A particle bouncing randomly in the small vessel tends to move to the large one. We may say that the space of states in the large vessel is larger. The cross section for a reaction is a measure of how large is the hole with respect to the volume of the states in the vessel.
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