Safely shooting antimatter

Question

Lets say I have this nice gun which can shoot a certain amount of antimatter. After hearing answers about why that won't work, I added a magical coating to the projectile making sure the antimatter doesn't react upon firing or while hitting air molecules. I recently also learned that matter-antimatter-collision results in gamma bursts, which may be bad for health.

Since I have no idea about the mathematics behind the collision, my question is how much antimatter should be used as a projectile so that the shooter doesn't die of gamma ray overdose shortly after shooting? The difference to the gamma radiation of the target is interesting too. For now take normal humans as shooter and target.

EDIT: To clarify: The goal isn't to kill the target with gamma radiation. The question is how much antimatter I can take before the resulting reaction affects the shooter in a bad way. Gamma radiation is just the byproduct of the collision.

EDIT2: It seems I created some confusion because I revealed to much story background. That's a downer, but I'll keep it in mind for next time. So let's simply image younger brother (30-year-old worker) of the leading antimatter scientist standing in big brother's particle accelerator by accident, about $m$ meters away from the spot where the antimatter is going to hit the matter. How much antimatter will the worker survive?

And I'm very grateful to Michael and Zxu for taking me serious, you guys rock! :-)

Answer 1

Michael Karnerfors gets the gist correct. However, it is misleading to simply say the antimatter should be "twice as powerful", since we only care about the backlash before- how much energy the bullet has itself is entirely irrelevant.

A more realistic approach:

Let's say the shooter is an 80 kg man. The maximum dose for a radiation worker per year in a non-emergency setting is 50 mSv = 4 J of gamma radiation (for his mass). According to linear no-threshold one big dose is the same as a bunch of small doses so we can just have him shoot one bullet a year without violating EPA rules. The reaction occurs 10 m away from you, with no additional shielding besides the air. Shielding from the air is <10% anyways at 10m.

Assuming that all photons pass uniformly outwards from the source, you recieve $\frac{A}{4*pi*10^2}$ of the original source, about 1/1200 of the the radiation if your frontal surface area is 1 sq meter. The permissible radiation from the source is 4800 J, and as e=mc^2+momentum and stuff.

Therefore $4800/c^2=m=5*10^{-11}$ g which is 50 picograms (one 20th of a nanogram). This scales linearly so if you want to shoot 50 bullets you'd get something closer to Michael's answer.

Answer 2

$E=mc^2$

Einstein's famous equation for mass-energy equivalence.

Anti-matter that is allowed to react with matter will annihilate into pure energy according to this equation, where $c$ is the speed of light (in a vacuum) and $m$ is the total mass annihilated.

So let us say you shot someone with a plain old 9 x 19 mm parabellum, but made of anti-matter. How much energy would be released?

In the normal case, the muzzle energy of a 9mm would be about 570 to 670 Joule.

In the anti-matter case, the impact will release $2\times 0.00745 kg \times (300\times10^6m/s)^2$ which is about 1 340 TeraJoule (Tera = one thousand billion), which in turn is about 20 times the Hiroshima bomb, 30 kilotons, or a large tactical nuclear explosion. Clearly you do not want to use this handgun ever because you can never shoot the bullet far enough to escape the blast wave of this explosion.

In order to scale this back you need to reduce the amount of antimatter in this bullet by the same order as you wish to reduce the energy output. So let us say you wanted the matter/antimatter-effect of the bullet to be only twice as destructive as the normal bullet.

Well then you need to reduce the mass by that same ratio. Then...

$1 340 \times10^{12} J / 670 J = 20 000 \times 10^{9}$.

20 thousand billionths of 7.5 grams is 0.4 thousands of a nanogram.

Well we can probably scale this up a little more, so we should be able to get up to another 2.5 times that energy without too much ill effect.

So there you have it: you can bring one one-thousand of a nanogram of anti-matter per bullet.

Answer 3

You don't want antimatter. At all.

Radiation dose is reverse proportional to the distance, squared. Twice as far means ¼ dose. If you want to kill by the radiation, it has to be effective at two meters. You know, to be sure head got lethal dose even if you only hit foot. With pistol, ten meters distance seems not to be uncommon. So you have 5 times more distance to you, than the "instant death" distance. ¹/₂₅ of the "instant death" dose is still going to kill you, eventually. Or make you ill. Even lower doses would be fatal, if you want to shoot more than once.