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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 (keff<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 (rc).

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;

 

kWfleUj.png

 

Where rmin 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 dkCX75N.png

Since the process is adiabatic, we know the following;

 

dByaZ0I.png

 

Where Ecore 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

 

vcdM4VM.png

 

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);

 

Oh0qc0f.png

 

(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

 

Q4rYYJ3.png

 

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 tc, we get the following expression;

 

FvdjFt9.png

 

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;

 

LC69wRA.png

 

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

 

yhG2HUa.png

 

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

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

 

oTLbDP5.png

 

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

 

2CGW9qk.png

qP1aH2i.png

 

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

 

obMnby7.png

 

Setting keff to 1 and solving for r, we find that the critical radius rc 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 keff equation, determining ρ is fairly simple.

 

CPEIm6w.png

 

 

Since keff 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;

 

0A5pR9r.png

 

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

HffJSlr.png

 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

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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.

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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.

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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.

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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.

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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.

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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?

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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.

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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);

 

2CGW9qk.png

(the lowercase sigma is the cross section in barns)

 

 

qP1aH2i.png

 

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).

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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.

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      Every engine has a maximum speed at which it can turn.  Often, the engine is artificially governed to a maximum speed slightly less than what it is mechanically capable of in order to reduce wear.  Additionally, most piston engines develop their maximum power at slightly less than their maximum speed due to valve timing issues:
       

      A typical power/speed relationship for an Otto Cycle engine.  Otto Cycle engines are primitive devices that are only used when the Brayton Cycle Master Race is unavailable.
       
      Most tanks have predominantly or purely mechanical drivetrains, which exchange rotational speed for torque by easily measurable ratios.  The maximum rotational speed of the engine, multiplied by the gear ratio of the highest gear in the transmission multiplied by the gear ratio of the final drives multiplied by the circumference of the drive sprocket will equal the gear ratio limit of the tank.  The tank is unable to achieve higher speeds than the gear ratio limit because it physically cannot spin its tracks around any faster.
       
      Most spec sheets don't actually give out the transmission ratios in different gears, but such excessively detailed specification sheets are provided in Germany's Tiger Tanks by Hilary Doyle and Thomas Jentz.  The gear ratios, final drive ratios, and maximum engine RPM of the Tiger II are all provided, along with a handy table of the vehicle's maximum speed in each gear.  In eighth gear, the top speed is given as 41.5 KPH, but that is at an engine speed of 3000 RPM, and in reality the German tank engines were governed to less than that in order to conserve their service life.  At a more realistic 2500 RPM, the mighty Tiger II would have managed 34.6 KPH.
       
      In principle there are analogous limits for electrical and hydraulic drive components based on free speeds and stall torques, but they are a little more complicated to actually calculate.
       

      Part of the transmission from an M4 Sherman, picture from Jeeps_Guns_Tanks' great Sherman website
       
      The Power Limit
       
      So a Tiger II could totally go 34.6 KPH in combat, right?  Well, perhaps.  And by "perhaps," I mean "lolololololol, fuck no."  I defy you to find me a test report where anybody manages to get a Tiger II over 33 KPH.  While the meticulous engineers of Henschel did accurately transcribe the gear ratios of the transmission and final drive accurately, and did manage to use their tape measures correctly when measuring the drive sprockets, their rosy projections of the top speed did not account for the power limit.
       
      As a tank moves, power from the engine is wasted in various ways and so is unavailable to accelerate the tank.  As the tank goes faster and faster, the magnitude of these power-wasting phenomena grows, until there is no surplus power to accelerate the tank any more.  The system reaches equilibrium, and the tank maxes out at some top speed where it hits its power limit (unless, of course, the tank hits its gear ratio limit first).
       
      The actual power available to a tank is not the same as the gross power of the motor.  Some of the gross horsepower of the motor has to be directed to fans to cool the engine (except, of course, in the case of the Brayton Cycle Master Race, whose engines are almost completely self-cooling).  The transmission and final drives are not perfectly efficient either, and waste a significant amount of the power flowing through them as heat.  As a result of this, the actual power available at the sprocket is typically between 61% and 74% of the engine's quoted gross power.
       
      Once the power does hit the drive sprocket, it is wasted in overcoming the friction of the tank's tracks, in churning up the ground the tank is on, and in aerodynamic drag.  I have helpfully listed these in the order of decreasing importance.
       
      The drag coefficient of a cube (which is a sufficiently accurate physical representation of a Tiger II) is .8. This, multiplied by half the fluid density of air (1.2 kg/m^3) times the velocity (9.4 m/s) squared times a rough frontal area of 3.8 by 3 meters gives a force of 483 newtons of drag.  This multiplied by the velocity of the tiger II gives 4.5 kilowatts, or about six horsepower lost to drag.  With the governor installed, the HL 230 could put out about 580 horsepower, which would be four hundred something horses at the sprocket, so the aerodynamic drag would be 1.5% of the total available power.  Negligible.  Tanks are just too slow to lose much power to aerodynamic effects.
       
      Losses to the soil can be important, depending on the surface the tank is operating on.  On a nice, hard surface like a paved road there will be minimal losses between the tank's tracks and the surface.  Off-road, however, the tank's tracks will start to sink into soil or mud, and more power will be wasted in churning up the soil.  If the soil is loose or boggy enough, the tank will simply sink in and be immobilized.  Tanks that spread their weight out over a larger area will lose less power, and be able to traverse soft soils at higher speed.  This paper from the UK shows the relationship between mean maximum pressure (MMP), and the increase in rolling resistance on various soils and sands in excruciating detail.  In general, tanks with more track area, with more and bigger road wheels, and with longer track pitch will have lower MMP, and will sink into soft soils less and therefore lose less top speed.
       
      The largest loss of power usually comes from friction within the tracks themselves.  This is sometimes called rolling resistance, but this term is also used to mean other, subtly different things, so it pays to be precise.  Compared to wheeled vehicles, tracked vehicles have extremely high rolling resistance, and lose a lot of power just keeping the tracks turning.  Rolling resistance is generally expressed as a dimensionless coefficient, CR, which multiplied against vehicle weight gives the force of friction.  This chart from R.M. Ogorkiewicz' Technology of Tanks shows experimentally determined rolling resistance coefficients for various tracked vehicles:
       

       
      The rolling resistance coefficients given here show that a tracked vehicle going on ideal testing ground conditions is about as efficient as a car driving over loose gravel.  It also shows that the rolling resistance increases with vehicle speed.  A rough approximation of this increase in CR is given by the equation CR=A+BV, where A and B are constants and V is vehicle speed.  Ogorkiewicz explains:
       
       
      It should be noted that the lubricated needle bearing track joints of which he speaks were only ever used by the Germans in WWII because they were insanely complicated.  Band tracks have lower rolling resistance than metal link tracks, but they really aren't practical for vehicles much above thirty tonnes.  Other ways of reducing rolling resistance include using larger road wheels, omitting return rollers, and reducing track tension.  Obviously, there are practical limits to these approaches.
       
      To calculate power losses due to rolling resistance, multiply vehicle weight by CR by vehicle velocity to get power lost.  The velocity at which the power lost to rolling resistance equals the power available at the sprocket is the power limit on the speed of the tank.
       
      The Suspension Limit
       
      The suspension limit on speed is starting to get dangerously far away from the world of spherical, frictionless horses where everything is easy to calculate using simple algebra, so I will be brief.  In addition to the continents of the world not being completely comprised of paved surfaces that minimize rolling resistance, the continents of the world are also not perfectly flat.  This means that in order to travel at high speed off road, tanks require some sort of suspension or else they would shake their crews into jelly.  If the crew is being shaken too much to operate effectively, then it doesn't really matter if a tank has a high enough gear ratio limit or power limit to go faster.  This is also particularly obnoxious because suspension performance is difficult to quantify, as it involves resonance frequencies, damping coefficients, and a bunch of other complicated shit.
       
      Suffice it to say, then, that a very rough estimate of the ride-smoothing qualities of a tank's suspension can be made from the total travel of its road wheels:
       

       
      This chart from Technology of Tanks is helpful.  A more detailed discussion of the subject of tank suspension can be found here.
       
      The Real World Rudely Intrudes
       
      So, how useful is high top speed in a tank in messy, hard-to-mathematically-express reality?  The answer might surprise you!
       

      A Wehrmacht M.A.N. combustotron Ausf G
       
      We'll take some whacks at everyone's favorite whipping boy; the Panther.
       
      A US report on a captured Panther Ausf G gives its top speed on roads as an absolutely blistering 60 KPH on roads.  The Soviets could only get their captured Ausf D to do 50 KPH, but compared to a Sherman, which is generally only credited with 40 KPH on roads, that's alarmingly fast.
       
      So, would this mean that the Panther enjoyed a mobility advantage over the Sherman?  Would this mean that it was better able to make quick advances and daring flanking maneuvers during a battle?
       
      No.
       
      In field tests the British found the panther to have lower off-road speed than a Churchill VII (the panther had a slightly busted transmission though).  In the same American report that credits the Panther Ausf G with a 60 KPH top speed on roads, it was found that off road the panther was almost exactly as fast as an M4A376W, with individual Shermans slightly outpacing the big cat or lagging behind it slightly.  Another US report from January 1945 states that over courses with many turns and curves, the Sherman would pull out ahead because the Sherman lost less speed negotiating corners.  Clearly, the Panther's advantage in straight line speed did not translate into better mobility in any combat scenario that did not involve drag racing.
       
      So what was going on with the Panther?  How could it leave everything but light tanks in the dust on a straight highway, but be outpaced by the ponderous Churchill heavy tank in actual field tests?
       

      Panther Ausf A tanks captured by the Soviets
       
      A British report from 1946 on the Panther's transmission explains what's going on.  The Panther's transmission had seven forward gears, but off-road it really couldn't make it out of fifth.  In other words, the Panther had an extremely high gear ratio limit that allowed it exceptional speed on roads.  However, the Panther's mediocre power to weight ratio (nominally 13 hp/ton for the RPM limited HL 230) meant that once the tank was off road and fighting mud, it only had a mediocre power limit.  Indeed, it is a testament to the efficiency of the Panther's running gear that it could keep up with Shermans at all, since the Panther's power to weight ratio was about 20% lower than that particular variant of Sherman.
       
      There were other factors limiting the Panther's speed in practical circumstances.  The geared steering system used in the Panther had different steering radii based on what gear the Panther was in.  The higher the gear, the wider the turn.  In theory this was excellent, but in practice the designers chose too wide a turn radius for each gear, which meant that for any but the gentlest turns the Panther's drive would need to slow down and downshift in order to complete the turn, thus sacrificing any speed advantage his tank enjoyed.
       
      So why would a tank be designed in such a strange fashion?  The British thought that the Panther was originally designed to be much lighter, and that the transmission had never been re-designed in order to compensate.  Given the weight gain that the Panther experienced early in development, this explanation seems like it may be partially true.  However, when interrogated, Ernst Kniepkamp, a senior engineer in Germany's wartime tank development bureaucracy, stated that the additional gears were there simply to give the Panther a high speed on roads, because it looked good to senior generals.
       
      So, this is the danger in evaluating tanks based on extremely simplistic performance metrics that look good on paper.  They may be simple to digest and simple to calculate, but in the messy real world, they may mean simply nothing.
    • By LostCosmonaut
      An interesting article about the natural fission reactors in Oklo, Gabon;
      http://www.scientificamerican.com/article/ancient-nuclear-reactor/
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