Nuclear disarmament verification via resonant phenomena

Nuclear disarmament treaties are not sufficient in and of themselves to neutralize the existential threat of the nuclear weapons. Technologies are necessary for verifying the authenticity of the nuclear warheads undergoing dismantlement before counting them toward a treaty partner’s obligation. Here we present a concept that leverages isotope-specific nuclear resonance phenomena to authenticate a warhead’s fissile components by comparing them to a previously authenticated template. All information is encrypted in the physical domain in a manner that amounts to a physical zero-knowledge proof system. Using Monte Carlo simulations, the system is shown to reveal no isotopic or geometric information about the weapon, while readily detecting hoaxing attempts. This nuclear technique can dramatically increase the reach and trustworthiness of future nuclear disarmament treaties.

TOF methods. The technique described in this work requires one to determine the energy of every neutron count in the detector. For the neutrons in the cold, thermal, and epithermal range this can be achieved via a pulsed source and time-of-flight (TOF) technique. If the neutron pulse occurs at time t 0 , and the non-relativistic neutron is detected at time t = t 0 + ∆t at a distance d then its energy is where m is the neutron mass and ∆t = t − t 0 is TOF. By propagating the errors we can determine the uncertainty in E: δE = δ∆tml 2 /∆t 3 . The uncertainty in ∆t primarily comes from that of t 0 , since most scintillation and microchannel based detectors have extremely high rise times. Here the uncertainty on t 0 is either the opening time of the chopper, or the pulse length of the accelerator that produces the epithermal neutrons via nuclear reactions. Thus, we can write Taking 5 eV as a point midway in our energy range, it is possible to determine the maximum time-width of the pulse to achieve the energy resolution of δE = 0.3 eV at the distance of l =5 m: δt 0 = ∆t 2 δE E . For this distance ∆t = 161µs, and thus δt 0 = 5µs. The precision of energy reconstruction can be increased by either making the pulse shorter, or by moving the detector further away and thus increasing ∆t. Since the geometric acceptance changes quadratically with l, it is statistically more optimal to shorten the pulse length than to lengthen the distance by the same fraction.
The TOF technique simply needs a neutron beam in a pulsed mode. In Supplementary Note 2 a few methods for such beams are described. For the platforms which use a DC neutron source, for example a reactor or a moderated isotopic neutron source, a chopper is necessary for producing a neutron pulse with a well defined t 0 . For most thermal beams cadmium is the preferred material, due to the ∼ 10 4 barns cross section of 113 Cd isotope. For the epithermal range of ∼5eV however the cross sections drop to just a few barns. A much better material for stopping the epithermal neutrons are boron and lithium. 10 B in particular, which has 25% abundance, has a cross section of approximately 250 barns. A boron chopper of thickness of just 3 mm will attenuate the neutrons by a factor of ×10 6 .
The chopper needs to be rotated at a frequency such that the opening time of its slot is ≤ 5µs, as determined above. Thus, a chopper could be a disk of radius of 50cm, made of 3mm of boron carbide sandwiched between two disks of steel of identical radius, for mechanical stability. If rotated at the frequency of 3600 rpm, a slit of 0.5 mm will yield an opening time of 2.7 µs. The chopper may have ten such equidistant slits, to recover some of the duty factor. Other possibilities may involve the use of turbomolecular pumps, which can easily achieve rotational speeds of 90000 rpm, with a narrow hole drilled through the boron-coated rotor blades. The neutral particle analyzer, built and installed at MIT's Alcator C-mod tokamak, could be an example of such a chopper 1 .
Epithermal neutron production by nuclear reactions and nuclear reactors. The epithermal neutrons can also be produced using nuclear reactions between accelerated light ions and various targets. There are two classes of light ion based nuclear reactions that can produce neutrons in the epithermal range. Significant work using epithermal neutrons was performed using the Los Alamos National Laboratory's 800 MeV proton spallation source, which produces neutrons of a broad range of energies 3 . This source was the basis of a number of beam lines used for a range of applied and fundamental studies. These included epithermal beams used for the non-destructive assay of nuclear fuels 4 . However a particularly attractive and compact alternative to a large spallation facility are smaller proton accelerators, which allow to trigger the such as 9 Be(p,n) 9 B. Some of these may have higher epithermal neutron yields. The search for an optimal reaction is outside of the scope of this work and may be a subject of future studies.
A significant challenge when using TOF techniques is the presence of the thermal neutrons, which arrive at times much longer than the waiting time for the epithermal pulse, thus making it difficult or impossible to identify the originating pulse. This could introduce uncertainties in the TOF reconstruction. However, the calculations shown in Supplementary Figure 1 show that the (p, n) reactions have almost no thermal flux, thus significantly limiting their impact on the precision of the TOF method.

Supplementary Note 3
Alignment, variations in design, and other systematic effects. In order to perform the verification measurement in a manner which is information secure, the hosts need to align the objects -the template/candidate with the reciprocal -in a way that the inspectors cannot observe the objects.
At the same time the inspectors need to be able to confirm visually that the boxes containing the template and candidate were switched between the two measurements. To achieve these dual goals the hosts need to place the three objects in three opaque boxes while aligning the objects with fiducial marks on the external surfaces of the box. Then, after the boxes are closed the inspectors can visually access the area. At this stage the template-reciprocal alignment can be performed using the fiducial marks. After the first measurements the template/candidate boxes are switched, and a new candidate-reciprocal alignment is performed. In this process the inspectors will not be able to observe the objects directly, the hosts will be able to perform the object alignments via the external fiducials, and the inspectors will be able to visually confirm that the template and the candidate have been switched between measurements.
Two issues come to the fore when discussing the use of reciprocals for a zero-knowledge proof system. As in any detection system, this verification system's sensitivity will have its limits, affecting the inspector's ability to distinguish between objects of various sizes or different isotopic concentrations. The detection probability of the system is determined by measurement times, detector sensitivities, as well as the specificity of neutron interaction physics. These factors are mostly of stochastic nature, and measurements of arbitrary sensitivity can theoretically be achieved by varying the measurement times. However, systematic effects are also present. These include the unit-to-unit variability, due to manufacturing precision, as well as the hosts' ability to align the template and the candidate with the reciprocal. Modern surveying methods allow alignment precision down to the O(10µm) scale, thus the main difficulty is related to the actual unit-to-unit variability.
Information on manufacturing precision are not available in open domain. For a given variability, the two sides can agree to broadened criteria of verification in order to accommodate such variability and thus avoid false alarms and reveal information about the geometry of the pit.
This circumstance in its turn limits one's ability to achieve arbitrary sensitivity. It is reasonable to assume that manufacturing variations are small, and thus the hoaxing scenarios attainable due to this limit on sensitivity are probably not of a significant advantage to either side in the inspection regime. A more rigorous treatment of this problem should be part of future research, possibly in the classified domain.

Supplementary Note 4
Probabilistic Tests and minimum necessary counts for 5σ detection. For a particular energy bin i, the statistical significance in units of sigma can be determined via assuming Poisson statistics, where c 0 and c 1 are the counts from the two distributions undergoing comparison. In frequentist statistical analysis, and assuming normally distributed errors we can determine the probability that the disagreement between c 0,i and c 1,i is consistent with the null hypothesis, i.e. is caused purely by statistical fluctuations. Conversely, the confidence for rejecting the null hypothesis and accepting the anomaly hypothesis (for example a hoaxing scenario is underway) can be determined via For example, an n = 5(σ) outcome (used in high energy physics for identifying new particles) corresponds to a confidence level of p = 1 − C(0, 5) = 1 − 2.9 × 10 −7 , a very high confidence that can be used as the standard of testing.
For a multi-bin data the more common test is the chi-square test. If the data has N bins, the number of degrees of freedom (NDF) is N. The probability that two distributions are deviating only due to normal fluctuations can be determined from p = P rob(χ 2 , N DF ), where P rob(x, y) is the chi-square distribution and χ 2 is the (non-reduced) chi-square that can be computed from If the object consists of a pit-reciprocal combination of thickness x and a vector r i , an extension plate (see the main body for an explanation) of thickness y and a vector r i , then the logarithm of total attenuation is simply where z i = (xr i + yr i )/(x + y) is the effective isotopic concentration vector of the combined pitreciprocal-extension. It can be shown that infinite combinations of x, y, r i , and r i will produce the same value of z i . To illustrate this consider three scenarios for r i , r i , and x = 5cm and y = 2cm: Here r max and r min correspond to all possible enrichment levels from which the extension can be made. The maximum is then just the super-grade plutonium, while the minimum can be the reactor grade plutonium. The full range of values of r is then simply By using the values of x = 5 cm, y = 2 cm, and solving for the 239 P u enrichment r 239 , we find that the range corresponds to about ∆r 238 = 23%, which is consistent with the previous result of 70-93%. This range can be further widened by either increasing y, or using r min of even lower enrichment.
As already stated, the simple calculation doesn't take into account such effects as in-scatter by neutrons. This necessitates a more thorough MC simulation to fully validate this idea. Such a simulation was performed using the MCNP5 package, and the results can be seen in Supplementary Figure 6 in the main body. The simulations confirm the conclusion of the analytic calculations above.
The importance of the above treatment is great: while the inspectors can use the data to reconstruct z i , they will not be able to reconstruct r i beyond simply stating that the pit enrichment level is somewhere between 70% and 93%. The knowledge of this broad range is essentially useless information, as it is already known that the plutonium in most weapons is at the WGPu enrichment levels. This range can be further broadened, if necessary. As discussed above that can be achieved either by using an extension of lower enrichment level or one of a thicker value of y -albeit at the need for longer measurement times. Finally, the reciprocal mask itself can be made modular: the recessed area shadowing the pit can be made of r i , while the peripheral part can be made from z i -thus removing the need for an extension plate.
While the analysis above shows that it is possible to protect the absolute isotopic information, some information about pit-to-pit variability may be inferred by the inspectors from comparative analysis of transmission spectra, for example by observing the variability in the absorption lines due to variable concentrations of 241 Pu, which has strong resonances at 4.2 and 8.5 eV. To mitigate this, the hosts and the inspectors could agree to a reduced resolution, as a way of smearing the absorption lines from that particular isotope. There are a few ways of achieving this. One approach would be to broaden the t 0 in the TOF technique by using a broader proton pulse for a 7 Li(p,n) 7 Be reaction 5 . For the case of a chopper technique a wider slit can be used.
If necessary, the information security of the system can be further strengthened by extending the the epithermal neutron source in this proof system with a velocity selection. Velocity selection is a well-established technique for filtering out neutrons based on their energy/velocity. A velocity selector is a system of multiple blades whose length, pitch angle and angular velocity allows only neutrons of a particular velocity range to pass through 10 . A yet simpler configuration would consist of two choppers: the first one setting the t 0 , and the second one, with a phase shift, selecting the neutrons based on their arrival time and thus their energy. Such a device could serve as a physical information barrier, allowing the hosts to limit the measurement to a particular pre-negotiated spectral region(s). Meanwhile the inspector can measure the velocity explicitly via the TOF information, as a way of confirming that the prover is not manipulating the output window of the velocity selector.

Supplementary Note 7
Reciprocal Geometries. The main function of the so-called reciprocal mask is to make it impossible for an observer to extract any sensitive isotopic or geometric information about the pit from a direct transmission measurement of the combined pit-reciprocal geometry. The simplest way of achieving this is by taking a space encompassed by a rectangular prism, filling it with a shape identical to the pit but with the negative of its density, then adding a uniform density until all the negative density voxels have zero density. This amounts to creating the negative of the pit. The 2-d cutaway of such a simple approach can be seen in Supplementary Figure 3.
While intuitively simple, this particular type of reciprocal mask has a number of problems.
For example, it would be very hard to keep subcritical. Even if the criticality of the mask can be significantly reduced (for example by slicing it perpendicular to the beam axis and introducing space between the slices), the combined thickness of pit-reciprocal configuration is unnecessarily high, thus necessitating long measurement times for a statistically significant detection.
A much more optimal reciprocal mask can be built simply by realizing that the thickness of the mask along a transmission axis needs to be equal to D − z, where D is some constant combined thickness and z is the thickness of the pit along that axis. So, for a hollow shell of internal and external radii r 1 and r 0 the reciprocal can be defined via its thickness along the beam axis d = D −2( r 2 0 − y 2 − r 2 1 − y 2 ) for y < r 1 and d = D −2 r 2 0 − y 2 for r 1 ≤ y ≤ r 0 , where y is the vertical coordinate. A combination of the pit and the reciprocal is illustrated in Figure 4.
For this particular case the combined thickness amounts to D = 5 cm.
The geometry and the enrichment of the combined pit and reciprocal is important when it comes to safety considerations. As suggested earlier, the wrong geometry may either be too close to criticality, or simply impossible to construct. Thus the criticality analysis of the geometries needs to be performed. As a neutron is incident on the pit or the reciprocal, it can trigger neutron We thus conclude that the impact of epithermalized fission neutrons from 240 Pu on the verification process will be negligible.

Supplementary Note 9
Epithermal Neutron Detectors. The warhead verification process described in this work uses epithermal neutron beams, which, after passing through the objects impinge upon a detector. The detector needs to allow the reconstruction of both the energy and hit coordinate of the incident neutron. The former is achieved via the TOF techniques described in Supplementary Note 1. To this effect the detector needs to have time resolution of less than a microsecond. To achieve a coordinate resolution it needs to be pixelated to a size agreed upon by the treaty parties.
A detector type which achieves both of these goals simultaneously is a 10 B doped microchannel plate (MCP), which has been developed by researchers from UC Berkeley and NOVA Scientific, Inc., and is described by Tremsin et al. 12 . The schematic in Supplementary Figure 5 illustrates the basic concept. The incident thermal or epithermal neutron is captured by the 10 B doppant in the glass, which leads to the reaction 10 B + n → 7 Li + α + 2.8M eV . The 2.8 MeV Q value of the reaction is shared between the alpha and the lithium ion, which then ionize the channel and start a cascade. The timing resolution of the MCPs is in the order of 100 ps. Comparing this to the δt = 5µs in Supplementary Note 1, we conclude that this timing resolution is abundant for achieving the energy resolution of δE ≤ 0.3 eV via TOF techniques. While the coordinate resolution of the MPCs may be excessively high, its readout may be modified as to sum multiple channel signals together, to achieve the desired resolution. The MCNP's precision could be limited by the cross section data in Evaluated Nuclear Data Files (ENDF), which it uses as input. To understand the scale of these possible discrepancies one can compare the nuclear cross section output from ENDF to the experimental data that they are based on. Figure 6 shows plots of ENDF cross sections along with experimental data for the four isotopes of plutonium, showing an overall good agreement.
To further strengthen the confidence in the ENDF, the experiments described in Losko et al. 4