Nuclear weapons were a radical improvement over previous weapons in terms of energy released per weight of explosive (relative effectiveness). They do not appear to represent a substantial discontinuity in terms of cost-effectiveness of explosives, though the evidence there is weak.

After the development of nuclear weapons, relative effectiveness improved quickly relative to historic rates. We do not have good enough data to assess later progress in explosive cost-effectiveness.

Background

The development of nuclear weapons is often referenced informally as an example of discontinuous technological progress. Discontinuities seem particularly plausible in this case to some observers because of the existence of a threshold phenomena in nuclear chain reactions.

Relative effectiveness of explosives

If we examine the maximum achievable density of explosives at any given point in history, we observe a sharp discontinuity at the point of nuclear weapons. The “relative effectiveness” of an explosive measures the amount of energy released per unit mass of the explosive, normalized so that the relative effectiveness of TNT is 1. The highest relative effectiveness of known explosives had grown to 1.7 by 1930. The relative effectiveness of an early nuclear weapon was around 4,500 (see Table 1).

 Technology Year developed  Relative effectiveness
 Gunpowder  800’s  0.5
 Nitroglycerin  1840’s  1.5
 Pentaerythritol tetranitrate (Penthrite)  1890’s  1.66
 Octogen  1930’s  1.7
“Fat man” (early atomic weapon).  1940’s  4,500
 B41 (most efficient nuclear weapon to date).  1960’s  5,100,000

Table 1: A rough timeline of explosive effectiveness. Derived from a timeline of explosives here and a comparison of explosive effectiveness here. These estimates modestly understate the impact of nuclear weapons, since the measured mass of the nuclear weapons includes the rest of the bomb while the conventional explosives are just for the explosive itself.

Table 1 elides many incremental improvements in explosive effectiveness. In particular we do not know whether early preparations of gunpowder were substantially less effective than modern preparations. However, Pentaerythritol tetranitrate does appear to be among the most powerful chemical explosives known, despite being only a small factor more effective than gunpowder. This appears to reflect fundamental limits on the energy density of chemical explosives. Once these limits were surpassed, attainable explosive densities increased rapidly, at first discontinuously and subsequently extremely quickly.

Relative effectiveness (RE) doubled less than twice in the 1100 years prior to nuclear weapons, then it doubled more than eleven times when the first nuclear weapons appeared. If we conservatively model previous progress as exponential, this is around 6000 years of progress in one step at previous rates.

Interestingly, at face value this discontinuous jump does not seem to be directly linked to the chain reaction that characterizes nuclear explosions, but rather to the massive gap between the energies involved in chemical interactions and nuclear interactions. It seems likely that similar results would obtain in other settings; for example, the accessible energy in nuclear fuel enormously exceeds the energy stored in chemical fuels, and so at some far future time we might expect a dramatic jump in the density with which we can store energy (though probably not the cost-effectiveness).

Cost-effectiveness of explosives

Background

Another important measure of progress in explosives is cost-effectiveness. Cost-effectiveness is particularly important to understand, because some plausible theories of continuous progress would predict continuous improvements in cost-effectiveness much more strongly than they would predict continuous improvements in explosive density.

Assessing the cost of nuclear weapons is not straightforward empirically, and depends on the measurement of cost. The development of nuclear weapons incurred a substantial upfront cost, and so for some time the average cost of nuclear weapons significantly exceeded their marginal cost. We provide estimates for the marginal costs of nuclear weapons, as well as for the “average” cost of all nuclear explosives produced by a certain date.

We focus our attention on WWII and the immediately following period, to understand the extent to which the development of nuclear weapons represented a discontinuous change in cost-effectiveness.

Cost-effectiveness of nuclear weapons

According to the Brookings Institute, nuclear weapons were by 1950 considered to be especially cost-effective, and adopted for this reason. Brookings say this has never been validated. This disagreement suggests that nuclear weapons are at least not radically more or less cost-effective than other weapons.

According to Wikipedia, the cost of the Manhattan project was about $26 billion (in 2014 dollars), 90% of which “was for building factories and producing the fissile materials.” The Brookings U.S. Nuclear Weapons Cost Study Project estimates the price as $20 billion 2014 dollars, resulting in similar conclusions. This post claims that 9 bombs were produced through the end of “Operation Crossroads” in 1946, citing Chuck Hansen’s Swords of Armageddon. The explosive power of these bombs was likely to be about 20kT, suggesting a total explosive capacity of 180kT. Anecdotes suggest that the cost to actually produce a bomb were about $25M, or about $335M in 2014 dollars. This would make the marginal cost around $16.8k per ton of TNT equivalent ($335M/20kT = $16.75k/T), and the average cost around $111k/T.

The US apparently plans to build 3,000 nuclear weapons for $60B. However it appears that at least some of these may be refurbishments rather than building from scratch, and the B61-12 design at least appears to be designed to be less powerful than it could be. The B61-12 is a 50kT weapon. These estimates give us $400/T ($60B/3,000*50kT). They are very approximate, for reasons given. However have not found better estimates. Note that they are not integral to our conclusions.

These estimates could likely be improved by a more careful survey, and extended to later nuclear weapons; the book Atomic Audit seems likely to contain useful resources.

Year  Description of explosive  Cost per ton TNT equivalent
 1920  Ammonium nitrate  $5.6k
 1920  TNT  $10.5k
 1946  9 (Mark 1 and Mark 3’s) x 20kt (marginal)  $16.8k (marginal Mark 3)
 1946  9 (Mark 1 and Mark 3’s) x 20kt (average)  $111k (average Mark 3)
 3,000 weapons in the 3+2 plan  $400

Figure 2: Total, average and marginal costs associated with different weapons arsenals

Cost-effectiveness of non-nuclear weapons

We have found little information about the cost of pre-nuclear bombs in the early 20th Century. However what we have suggests they cost a similar amount to nuclear weapons, for a certain amount of explosive energy.

Ammonium nitrate and TNT appear to be large components of many high explosives used in WWII. For instance, blockbuster bombs were apparently filled with amatol, which is a mixture of TNT and ammonium nitrate.

An appropriations bill from 1920 (p289) suggests that the 1920 price of ammonium nitrate was about $0.10-0.16 per pound, which is about $1.18 per pound in 2014. It suggests TNT was $0.44 per pound, or around $5.20 per pound in 2014. These estimates are consistent with that of a Quora commenter.

This puts TNT at $10.4k/ton: very close to the $16.8k/ton marginal cost of an equivalent energy from Mark 3 nuclear weapons, and well below the average cost of Mark 3 nuclear weapons produced by the end of Operation Crossroads.

Ammonium nitrate is about half as energy dense as TNT, suggesting a price of about $5.6k/T ($1.18 * 1/0.42 * 2000). This is substantially lower than the marginal cost of the Mark 3.

Note that these figures are for explosive material only, whereas the costs of nuclear weapons used here are more inclusive. Ammonium nitrate may be far from the most expensive component of amatol-based explosives, and so what we have may be a very substantial underestimate for the price of conventional explosives. There is also some error from synergy between the components of amatol.

Conclusion

Without a longer run price trend explosives, it is hard to say whether a development is surprisingly quick. However from the evidence we have here, it is unclear that nuclear weapons represent any development at all in cost-effectiveness, in terms of explosive power per dollar. Thus it seems unlikely that nuclear weapons were surprisingly cost-effective, at least on that metric.

Implications

The discontinuity in relative effectiveness produced by nuclear weapons is the largest discontinuity in any technological progress trend that we are aware of, as of June 19 2015. It is also one of very few large discontinuities that we know of. This makes it an important datapoint for learning about which processes do see large discontinuities, and which do not.1

  1. We have speculated about its causes on the blog, but await further data before investigating more seriously.