So You Want to Build a Fission Bomb

[LostCosmonaut]

So You Want to Build a Fission Bomb?

There are many reasons why one may want to build a fission bomb. Killing communists, for example, or sending a spacecraft to one of the outer planets. Building a bomb is not easy, but it can be done (see also Project, Manhattan). Even with publicly available information.

 

Obviously, I’m not going to detail every little bit of our hypothetical bomb down to the last millimeter of wiring. First, I don’t know all that. If I did know it, posting it here might earn me a very long vacation to ADX Florence. The stuff here is just some equations and such to give you a general impression of how the design looks.

The core of our hypothetical bomb is a sphere of highly enriched uranium. We want it to be subcritical (k_eff<1), but not by much. The more subcritical it is, the more we have to compress it to make it critical. In a real bomb, the core is usually surrounded by a layer of dense material such as tungsten or depleted uranium called the tamper. This helps keep the core together longer, and if it’s made of U-238, you can get some extra yield from the tamper fast fissioning. To simplify our analysis, our bomb won’t have a tamper. Then, you have a bunch of chemical explosives on the outside. This is what compresses the core, and takes it from subcritical to a super prompt critical state.

 

When the core is super prompt critical, it’s going to heat up very quickly. Within milliseconds, the uranium at the center is going to become hot enough to be a gas (at very high pressure). At the edge of the core, you’re going to have very high pressure uranium gas next to an area of very low pressure. This is going to result in the uranium gas blowing off very quickly. This results in a “rarefaction wave” forming, as the core progressively evaporates away. This rarefaction wave proceeds inward at the speed of sound, and once it gets far enough in, the core becomes subcritical, and the reaction stops.

Now, I’m going to make a few assumptions. These will result in some inaccuracy in our calculations, but the results will be close enough (also, it makes everything much simpler). Here they are;

 

1. The super prompt critical condition of the core will terminate once the rarefaction wave reaches the critical radius (r_c).

2. The super prompt critical reactivity will remain constant until the core is subcritical.

3. The core is spherical with no tamper.

4. The temperature of the core is high enough that it can be treated as a photon gas (radiation pressure is the dominant force.)

5. No energy is lost to the surroundings during the process (adiabatic).

6. Our core is made of pure U-235.

 

Since the rarefaction wave proceeds inward at the speed of sound, the device is critical for the following period of time;

 

 

Where r_min is the radius of the core at maximum compression, and a is the speed of sound. We’ll also assume the gaseous core has a specific heat ratio of 4/3, so . 

Since the process is adiabatic, we know the following;

 

 

Where E_core is internal energy of the core at the end of the period of prompt criticality (this is the amount of energy released in the detonation). Substitute that into the speed of sound equation, and we get

 

 

Putting that aside for a moment, let’s take a look at the point kinetics equation, which describes how power increases in a reactor following a sudden increase in reactivity (our bomb is essentially a reactor that’s undergone a massive increase in reactivity);

 

 

(In this case, ρ represents reactivity, instead of density. β is the fraction of fission decay products which decay through neutron emission, and Λ is the average prompt neutron lifetime.)

The second term in that equation gives us the power contribution from delayed neutrons, so we can ignore in this case (the bomb will have long since detonated by the time they become a factor). Also, in the case of super prompt criticality, ρ >> β. So our equation reduces to

 

 

So to get the total amount of energy produced in the core during super prompt criticality, we need to integrate the power equation over the amount of time the core is super prompt critical. If we call that time t_c, we get the following expression;

 

 

Where E₁ is the amount of energy produced by one fission event (202.5 MeV). Substituting that into our first equation and the speed of sound expression, and then doing a bit of algebra (which I’ll leave out for the sake of brevity), we end up with this;

 

 

Solving for the E_core expression, and defining Δr as the difference between criticality radius and the radius at maximum compression;

 

 

Which gives us the total amount of energy released by the detonation.

The main unknowns here are the reactivity (ρ) and critical radius (r_c). Fortunately, both of these are fairly easy to determine. The critical radius is the radius at which a sphere of material has a k_eff (ratio of neutron production to neutron absorption) of 1.

 

 

ν is the average amount of neutrons produced per fission event (~2.5), Σ_f is the fission cross section (σ_f = ~1 barn for fast neutrons), D is the thermal neutron diffusion distance (.00434m for U-235), B_g is the ‘geometric buckling’, and Σ_a is the absorption cross section (σ_a=~.09 barns for fast neutrons). Convert from σ_f to Σ_f using the following formulas;

 

 

B_g for a sphere can be calculated using the following formula;

 

 

Setting k_eff to 1 and solving for r, we find that the critical radius r_c is roughly 5.2cm. A sphere of U-235 of this size will have a mass of about 11.25kg.

Now that we have the k_eff equation, determining ρ is fairly simple.

 

 

 

Since k_eff is going to be higher the more you compress the core, you obviously want to compress it as much as possible. The following equation gives the amount of explosive needed to compress the core by a given amount;

 

 

E_scf_c is the amount of energy needed to compress the core by a given amount.

 η is the amount of energy contained in each unit of chemical explosive (4184kJ/kg for TNT), and ε is the efficiency of the implosion process. ε is about 30% in well-designed nuclear weapons, crude designs are probably closer to 5-10%.

 

 

Congratulations! Now you have (a non-trivial portion of) the knowledge you need to build a working fission device!

 

Edit: Updated 4/24 with corrected cross sections

[Toxn]

Supercriticality was touched upon recently here. It's the for-dummies version, but still.

[Collimatrix]

That's a very good link, Toxn.  Thanks for sharing it.

 

We need a good explanation of the difference between critical and prompt critical.  Obviously, Unstart knows what the difference is, and I know what the difference is, but the precise explanation is in the equations above rather than spelled out in words.

 

It'll be in the next section of my guide to nuclear energy, but Unstart might take a whack at it first.

 

Thanks for writing this, Unstart.  Really top notch stuff.

[LostCosmonaut]

Technical definition of criticalilty: the ratio of neutron production to neutron absorbption is 1.

 

Useful definition: A critical reactor is operating at a steady power level. If the amount of neutrons you have in your system is constant, you're going to have the same rate of fission events happening (fission events are how you get energy), so your power level will be constant.

 

Supercritical: You're producing more neutrons than are being absorbed. So, you're going to have more fission events happening, so your power level will increase.

 

Prompt Critical: Your ratio of neutron production to neutron absorption is very very high (>1.00065, according to my notes). Your power level is going to increase exponentially. At this point, virtually all of your power is coming from prompt neutrons, and delayed neutrons have almost no effect.

 

Most reactors don't stay at prompt critical for more than a very small fraction of a second. This is because a prompt critical reactor produces obscene amounts of heat, which tends to either melt the reactor or blow it apart entirely. Either of these events results in the system reaching a subcritical geometry.

[Collimatrix]

My understanding is that prompt criticality is when the production of prompt neutrons divided by the absorption of neutrons is >1.  Since prompt neutrons are produced more or less instantaneously with each fission event (thus the name), a super prompt critical configuration will experience exponential increases in reactivity very quickly.

 

You can have a super-critical configuration that isn't prompt critical, such as a nuclear reactor increasing its core temperature.  It will experience exponential increases in reactivity, but slowly.

 

At which point a bunch of hippies write a post-structuralist critique of nuclear energy in a journal called Critical Theory.

[LostCosmonaut]

Yep, that's about the sum of it. A normally operating reactor will be subcritical based on prompt neutrons, but the delayed neutrons push it over the edge. Since the delayed neutrons are produced on timescales on the order of minutes, they can be controlled.

For an example of a prompt critical situation, SL-1 went from 3MW to 20000MW in less than a second. And that was in error, rather than by design. When you do intentionally, you end up with Robert Oppenheimer referencing Hindu scripture.

[Collimatrix]

Oh good, we're on the same page then.

 

Is there a reason you're using the thermal cross section for sigma sub a?  I thought nuclear weapons, barring the unsuccessful uranium hydride designs, used strictly fast neutrons?

Also, I was sad when I saw that all the equations are images.  I thought for a second that this forum supported latex.

[Toxn]

How possible is it to brute-force the reaction by going prompt critical inside a really heftily-constructed housing or something?

 

I mean, SL-1 disassembled itself (and its three operators) in something like 10 milliseconds. So what would have happened if it had been operating encased in a big ol' chunk of steel designed to keep the reactor together as long as possible?

[Collimatrix]

Well-designed reactors typically have passive means of preventing sudden excursions in core power.

 

The most common power-generating reactors are light water and boiling water reactors.  These use regular, every day water as a neutron moderator and as a working fluid/coolant in the turbines.

 

Because of reasons, low-velocity neutrons are more likely to cause fission in uranium-235.  Couldn't tell you exactly why; something about resonance.

 

Regular, every day water turns out to be a pretty good neutron slower-downer ("moderator").

 

When the core starts to heat up, more of the water boils.  As the water boils, its density goes down and it becomes a less effective moderator, which means that fewer of the neutrons cause fission.  Any sudden excursions in power are self-limiting.

 

This is called a negative void coefficient of reactivity, but there are other passive ways of keeping things from getting out of hand and exploding.

 

 

 

Edit:

More to the point, in a well-designed reactor it shouldn't be possible to reach a prompt super critical state in the first place at all!

 

Edit edit:

To answer your second question, Toxn, assuming you had a reactor that was designed by knotheads and they'd neglected to make it impossible for a prompt super critical state to occur, you could make it explode, but it would be a weak, inefficient explosion compared to a proper nuclear weapon.

 

As Unstart described, in a prompt super critical event, the absurd amount of energy being released in less than the blink of an eye tears apart the fissile material and turns it into plasma.  Plasma isn't dense enough to be critical anymore, so the trick in designing these things is to try and get as much fission to occur before the device deconstructs itself (and everything in a several kilometer radius).

 

There are all sorts of tricks that are required to get lots of fission to happen in a short time, and to put it flippantly, reactors don't have these tricks because they're not designed to explode.  These are tricks like modulated neutron initiators, neutron reflectors, fusion boosting, and probably a bunch of stuff that's still secret.  Basically you're trying to make the curve of the exponential increase in reactivity as steep as possible, and lop off the left part of the curve as much as possible.

[Toxn]

I'm not talking about a well-designed reactor.

[LostCosmonaut]

Oh good, we're on the same page then.

 

Is there a reason you're using the thermal cross section for sigma sub a?  I thought nuclear weapons, barring the unsuccessful uranium hydride designs, used strictly fast neutrons?

Also, I was sad when I saw that all the equations are images.  I thought for a second that this forum supported latex.

 

That's my bad, I got the values out of my notes, which happened to be a bit where they were discussing reactors, so the value listed was for slow neutrons. This site will give you the cross sections of various nuclides at various energies, and is almost certainly more accurate than my notes. I've replaced the cross sections with those for fast neutrons, and redid my calcs (turns out I was off on the critical radius by about .5cm).

It would indeed be quite nice if this site had some sort of equation editor.

 

 

Edit: One thing which I critically forgot to mention is that the cross sections aren't usually left in barns, but converted to units of 1/cm (or cm²/cm³ if you prefer);

 

(the lowercase sigma is the cross section in barns)

 

 

 

I've gone and edited these into the OP, as well as updating my calcs to make them more accurate.

I'm not talking about a well-designed reactor.

 

Even in such a shittly designed reactor such as the RBMK, it took quite a bit of idiotic fuckery to get it prompt critical. If you absolutely need to get your fissiles prompt critical, I recommend precise application of explosives (or, for advanced users, radiation pressure).

[Collimatrix]

I'm not talking about a well-designed reactor.

 

OK, gotcha.  Worst-case scenario with a reactor designed by complete drooling idiots.  No passive negative feedback loops, and no prevention of a prompt super critical core configuration.

 

It'll explode, but relatively little of the fissile material will fission.  By bomb standards it's a fizzle.  It will still be a mighty explosion by non-nuclear standards, but even with a bunch of steel acting as a tamper as you suggested, it will be a very inefficient bomb.

 

In a bomb, you try to prevent any fission from happening until the core is completely collapsed on itself.  Since the collapsing core is in a super prompt critical state, any fission reactions will self-multiply and the device will self-disassemble before most of the material fissions.  Sexual metaphors apply.

 

So instead, bombs are designed to reach maximum compressed density, and then start fissioning as quickly as possible.  A little device called a modulated neutron initiator squirts a bunch of neutrons into the bomb core as soon as the implosion crushes it, and the fission gets going as crisply and efficiently as possible.  This is also why plutonium from a typical reactor is garbage for bombs; it has too high of a spontaneous fission rate, and it goes super prompt critical prematurely.  Instead of an earth-shattering kaboom there's a fizzle and a bunch of survivors and it's very embarrassing and the engineers swear that it never happened to them before.

 

In a reactor, there's fission going on all the time, so any rapid, super prompt critical excursions will happen without anything close to optimal fuel density.

 

So you'll get a big bang that breaks the reactor and probably anything in the immediate vicinity, and a good deal of radioactive nastiness in the air, but it won't flatten cities or anything.