# How does thermonuclear warhead yield scale with size?

The Tsar Bomba packed a yield of 50 megatons into a package 2.1m in diameter by 8m long. Assuming that the lead tamper modification was not used (which boosts the yield of the Tsar Bomba design to 100 megatons), how much would the yield scale as a multiple-stage thermonuclear warhead (assume that extra stages are added as-needed) was scaled up, say to 3.5m diameter and 20m length?

• You'd probably be looking at the square-cube law here.
– Monty Wild
Commented Feb 18, 2015 at 2:20
• @MontyWild -- I think I am, but in this case, wouldn't it play in my favor? (i.e. my thought is that yield is proportional to the volume of fusible material available)
– Shalvenay
Commented Feb 18, 2015 at 2:23
• While 'yield' in megatons may rise as cube-square, blast radius rises only with cube root of the yield. That's why MIRV etc are more efficient; 100MT of yield spread between 100 1MT bombs will cause much more destruction than a single 100MT bomb (unless you target very deep underground bunkers).
– SF.
Commented Feb 19, 2015 at 1:54

Disclaimer: I am not a nuclear engineer.

From what I've read, nuclear weapon design is a very complex science. I don't think that you could reliably describe a scaling rule. What you could do is look at some historical data and extrapolate from there. However, extrapolation is dangerous™ and you should not assume that something will continue to scale in the same way as it has for your existing data points.

Since we are (hopefully) unlikely to obtain empirical data from a 50+ megaton detonation, extrapolation is the best that we can do. Wikipedia has some information that we can use to perform these calculations. They do not list dimensions but they do list a number of masses. They also provide yield-per-mass numbers (6 MT/ton or 25 TJ/kg theoretical, 5.2 MT/ton practical). Based on the page you cited, the Tsar Bomba had a yield-per-mass of 8.89 TJ/kg. This is clearly significantly below the theoretical max. As with many engineering devices, it is likely that at the 50 MT scale you cannot achieve anything close to theoretical max efficiency.

If you assume that you can maintain this level of efficiency for a device of a greater size (and therefore a greater mass) you can multiply 8.89 times the mass in kg of the hypothetical device and this would give you a reasonable WAG concerning the yield. I would guess that your efficiency would continue to decline so you can account for that by rounding down your answer to make it a little more accurate.

Hope that helps!

I am also not a nuclear engineer, but here are my thoughts on it:

My impression is that you might have trouble scaling a weapon in an arbitrary dimension but that they should scale basically linearly with mass assuming an acceptable configuration of the weapon.

In practice you reach a maximum size for your h-bomb based on the input energy available and various mechanics that they try to keep secret. However, to detonate an h-bomb you simply need an adequate energy flux, the source of that energy flux is irrelevant. In practice it's an fission bomb because nothing less than this can provide the needed energy. However, there's nothing about the fission part of it that's actually needed--a fusion detonator works even better. (Besides, the fusion stage of an h-bomb actually contains a simplified fission device in it's heart--there is a rod of Pu-239 there, after the energy flux has crushed the sides of the fusion device in then the result gets crushed lengthwise, turning the rod into a critical mass and setting off the crushed lithium deutride.)

Thus you can simply stack up the fusion stages of h-bombs one after another, the energy flux from one sets off the next. You can also arrange multiple fusion devices around the fission trigger. Just watch your configuration that you don't destroy a stage before it's fired.