What would a Positron beam look like?

[Knife]

So I was thinking the other day. Star Trek relies on anti matter for it's engines and power and they tend to have a lot of it around, why do they not use them for weapons? Minus, of course, the photon torpedos. Then I got to thinking; what would a hit from a positron beam look like? Would you have like a chain of explosion type cascades as new antimatter hits new matter in the stream? Would it look more like an eating away of the surface of the target? What kind of radiation and thermal effects would one see?

[Surlethe]

The positrons will annihilate with electrons, so you can regard this as a form of ionizing radiation coupled with gamma ray release. The heating will probably depend on how opaque the struck material is to gamma rays (specifically, 512 keV gamma rays). How far it penetrates depends on the energy of the positron beam; if I have my understanding correct, a very high-energy beam would pass through the material with few to no collisions; a low-energy beam would not penetrate very far.

This is interesting me; here is a quick heuristic model. The beam, with intensity I, will attenuate at a rate that depends on the density of the matter being penetrated. Therefore, I is a function of density p and distance z. Gamma ray creation z from the surface depends on the rate at which the beam attenuates, dI/dz. Model this as a cylindrical emission of gamma rays; it should be radially symmetric, so it's a function only of z. The heating of the material will depend on the rate at which it absorbs gamma rays, which is a function of r, distance from the beam. You also need to take into effect the ionization effects, but I'm not sure how to do that; you'll have heavy positive ion cores bouncing around. They'll repel each other and expand, adding an additional heating effect as they collide with other (neutral) atoms and spread the kinetic energy around.

When I get home, if I remember & have time, I'll see if I can draw up a quick actual model of this.

[Kuroneko]

Unless the positrons are extremely relativistic, the reaction would not be energetic enough to create any more than trace amounts of secondary antimatter. Electron-positron reaction is dominated by gamma radiation at 511keV. Since electron binding energies, whether due to intermolecular bonds or ionization, are significantly less than this, almost all of the initial reaction will be in this form. For reference:

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511keV     Air    Water  Iron    interpolation from L'Annunziata, _Radioactivity_
μ_m        0.086  0.096  0.097   total mass attenuation in cm²/g
HVT        6.7E3  7.2E0  0.91    half-value thickness in m

So far, it looks mostly to be a burst of gamma radiation of low interactivity, but things might not be that bad. A relativistic positron having Lorentz factor γ in the x direction encountering an electron of mass m will have a total four-momentum of p = [ -E/c; p; 0 ], p = γmv, E = (γ+1)mc². We know that energy and spatial momentum are conserved, so we can expect the average case to look like
[1] p = p'₊ + p'₋, p'_± = [ -E/2c ; p' cos θ ; ± p' sin θ ],
where θ is the opening half-angle of the products. Requiring p'_± to be lightlike imposes E/2 = p'c, giving the solution
[2] cos θ = γv/[c(γ+1)].
For example, at γ = 2 (not highly relativistic, but equal amounts of rest and kinetic energy), θ = 54.7° will be the opening half-angle for the average gamma ray products.

Thus, higher positron energies not only lead to tighter collimation of the beam itself, but also of the resulting gamma rays, but at a cost of lower interactivity, since this also means that the gammas will be more energetic than the minimum 511keV.

[Kuroneko]

Erratum: for 'average', read 'median' in the previous post. The half-opening angle found therein represents a cone such that half of the emitted photons fall within it. The power distribution, however, is a bit more complicated due to Doppler shift.

In spherical coordinates with respective azimuth and zenith angles θ and φ, consider a stationary particle of mass 2M decaying into two photons, each of energy mc² and opposite directions to conserve momentum. Assuming that the emission of one of the photons has a uniform probability distribution over the solid angles, the individual probability density functions are
[1] t(θ) = 1/(2π), f(φ) = (1/2)sin φ.
For a photon of energy mc² emitted in a particular direction, the four-momentum is
[2] p = Mc[ -1 ; cos θ sin φ ; sin θ sin φ ; cos φ ].
Performing a Lorentz boost to a frame in which the particle has a velocity v in the zenith direction gives
[3] p' = Mc[ -γ(1 + v/c cos φ) ; ... ; γ(cos φ + v/c) ],
so that only the photon's zenith angle is modified according to
[4] cos φ' = (cos φ + v/c)/(1 + v/c cos φ),
while changing the variables produces the probability density in the boosted frame:
[5] f'(φ') = (1/2)(1-v²)[sin φ']/[1 - 2v cos φ' + v² cos²φ' ].
This, from rest, the relative power density will be P(φ') = D(φ')f'(φ'), where D is the Doppler-shift factor (cf. [3])
[6] D = γ(1 + v/c cos φ) = 1/[γ(1-v/c cos φ')].

Summary:
(a) A positron (mass m) of some velocity V and Lorentz factor Γ = (1-V²)^{-1/2}, encounters a stationary electron.
(b) The center-of-mass frame between the two particles has velocity v and Lorentz factor γ = (1-v²)^{-1/2}, satisfying v = ΓV/[c²(Γ+1)]. In this frame, the total momentum of the two particles is zero, but the total effective mass is 2M = 2γm.
(c) In the COM frame, we assume that the electron-positron pair emits only gamma rays in a completely uniform manner, having total energy 2Mc². This is not quite true for high Lorentz factors, but other reactions have small amplitudes, so this ought to be good enough.
(d) In the rest frame, the total energy per reaction is γ(2M)c² = (Γ+1)mc², but relativistic aberration and Doppler shift makes the power distribution have a conal pattern.

Below is MATLAB output for the quartile half-opening angles based on positron beam Lorentz factors, in degrees. For example, at Γ = 10.0 (V = 0.995c), 50% of resulting energy will fall within φ'≤16.97°, 75% within φ'≤26.46°, and 99.34% within φ'≤90°, etc.

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Beam       Q1        Q2        Q3        90°frac
1.000    60.00     90.00    120.00       0.5000
1.500    32.62     52.91     79.96       0.8130
2.000    26.27     42.94     66.17       0.8849
2.500    22.67     37.13     57.63       0.9208
5.000    15.06     24.70     38.56       0.9763
7.500    12.06     19.78     30.87       0.9887
10.00    10.35     16.97     26.46       0.9934
20.00     7.22     11.83     18.42       0.9982
100.0     3.20      5.23      8.13       0.9999

Thus, the gamma burst will be focused in the direction of the beam, and for higher Lorentz factors practically all of the energy will be delivered inward. Beyond that, specific details about both the beam and the target are necessary to answer the original question.

[Surlethe]

Oh, I see - since momentum has to be conserved, the gamma ray photons need to be emitted chiefly in the direction of the positron beam.

What about the ionizing effects of the beam? If some of the material has all of its electrons turned to gamma rays shooting off elsewhere, the Coulombic repulsion will heat it.

[Kuroneko]

For breaking chemical bonds, you can look up the differences in enthalpy, latent heats, and the ionization energies of the atoms. For example, cumulative ionization energies for iron are

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Fe  1      2      3      4      5      6      7      8      9      10   electrons
   0.0079 0.0240 0.0547 0.1095 0.1846 0.2841 0.4095 0.5605 0.7956 1.0577 keV

Since this is still very low compared to a reasonable beam energies, the effect of this on the annihilation emission spectrum can be ignored with relative impunity. Post-annihilation, the effects of Coulomb repulsion can be modeled by an effective temperature corresponding to electrostatic potential energy between the resulting ions. What concerns me more is the ionization due to bremsstrahlung, which could potentially change the radiation spectrum significantly.

So before even we get to the secondary effects of the ions, we need to consider
(1) the deceleration of the positron while it travels through matter, its bremsstrahlung emission spectrum, and the associated ionization, and
(2) electron-positron annihilation after that deceleration phase (this is probabilistic), which has distribution as calculated in the previous post.
The annihilation cross-section for a positron in the 1-10MeV range on an iron atom is on the order of a barn, which corresponds to a mean free path in iron on the order of a centimeter. That's not much on the macro-scale, but a centimeter still takes it near a lot of other atoms, so the opportunity for interaction might be very significant.

I wish the OP was a bit more specific on the parameters of the scenario. We're not even clear on the positron energies involved or the target characteristics. I've been assuming MeV-range for the former--this being (Γ-1)(0.511MeV)--which is easily achievable now, but in sci-fi scenarios it may be reasonable to have GeV-range positrons (technically, we can push toward a TeV already, but the requisite accelerators are huge) in a small enough package. For the target, I've assumed the hull is mostly iron so far (but it may be something else, e.g., titanium or some composite material), of yet unspecified thickness and overall target size and density.

Note: in the previous post near the beginning, "each of energy mc²" should read "each of energy Mc²", since m and M are different quantities there. That should be clear from context, but I mention it just in case.

[Darth Wong]

This is what I love about Kuroneko: he actually does serious work for these questions. If we had a payroll, I'd put him on it

Anyway, it seems that the visual effect of the beam striking the target should be similar to that of a laser, at least to the naked eye. Pretty disappointing if an observer is expecting something unique and remarkable.

[Surlethe]

Post-annihilation, the effects of Coulomb repulsion can be modeled by an effective temperature corresponding to electrostatic potential energy between the resulting ions.

You mathematician - "A model exists, now on to other things!" Seriously, though, if the radiation resulting from the annihilation (chiefly high-energy gamma rays) does not interact significantly with the material, won't the effective temperature from the Coulomb repulsion be the most significant effect on the target material itself?

[Sky Captain]

Is it really worth to bother with positron or antiproton particle beams? As I understand if the particle beam is highly relativistic most of it`s destructive energy will come from relativistic mass increase of particles not from annihilation with matter.

[Knife]

I wish the OP was a bit more specific on the parameters of the scenario.

As far as weapons characteristics? I don't know. However for targets, I was actually a bit curious about a couple things. One; in atmosphere, would not any of the positively charged electrons react with matter in the air? Would you not get a 'detonation' as soon as the 'beam' left the emitter?

Anyway, in vacuum or close to it, I assume most targets would be primarily iron/steel but I would be interested in the effects of some composites. As far as density, I'm assuming and increase in input would equal a longer duration on target which would equal deeper penetration. I would also assume, based off of your earlier post, that the denser the 'armor' the higher chance of collision of the two particles to make a reaction.

[Surlethe]

I think the parameters he's referring to are beam intensity, beam collimation, average positron speed (related to beam intensity), and the properties of the material being impacted (density, atomic number, etc.).

[Knife]

I think the parameters he's referring to are beam intensity, beam collimation, average positron speed (related to beam intensity), and the properties of the material being impacted (density, atomic number, etc.).

Well he also stated something about target characteristics, but as to beam characteristics, I have no idea. As far as density; Fe is typically around 7800 kg/cu M. as is steel but I'm not sure what the common densities of of common composites are.

[Fingolfin_Noldor]

Kuroneko, can such a beam be treated as a wave in the paraxial approximation, like a Gaussian beam? Or are there other beam dynamics to consider given that positrons have charge?