Development of an isotope-sensitive warhead verification technique using nuclear resonance fluorescence

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Development of an isotope-sensitive warhead verification

technique using nuclear resonance fluorescence

by

Jayson Robert Vavrek

B.Sc. Physics, University of Alberta (2014)

S.M. Nuclear Science and Engineering, Massachusetts Institute of Technology (2016) Submitted to the Department of Nuclear Science and Engineering

in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Nuclear Science and Engineering

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019

Author . . . . Jayson Robert Vavrek Department of Nuclear Science and Engineering May 24, 2019 Certified by . . . .

Areg Danagoulian Assistant Professor of Nuclear Science and Engineering Thesis Supervisor Certified by . . . .

R. Scott Kemp MIT Class of ’43 Associate Professor of Nuclear Science and Engineering Thesis Reader Accepted by . . . . Ju Li Battelle Energy Alliance Professor of Nuclear Science and Engineering Professor of Materials Science and Engineering Chair, Department Committee on Graduate Students

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Development of an isotope-sensitive warhead verification technique using nuclear resonance fluorescence

by

Jayson Robert Vavrek

Submitted to the Department of Nuclear Science and Engineering on May 24, 2019, in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy in Nuclear Science and Engineering Abstract

Nearly three decades after the end of the Cold War, nuclear arms control remains a pressing global issue. Despite obligations under the Treaty on the Non-Proliferation of Nuclear Weapons (NPT) to make ‘good-faith efforts’ to disarm their nuclear arsenals, the United States and Russia still maintain thousands of nuclear warheads. Progress towards complete disarmament has been gradual due to a variety of sociopolitical barriers, but future efforts towards nuclear arms reduction will face an additional technological hurdle: no technology exists to verify that warheads slated for dismantlement are authentic without revealing any sensitive weapons design information in the process. Despite several decades of research, no technology has solved this apparent paradox between information security and confidence in a warhead verification measurement.

Recent work by Kemp, Danagoulian, Macdonald, and Vavrek [1] has produced a novel physical cryptographic verification protocol that approaches this treaty verification problem by exploiting the isotope-specific nature of nuclear resonance fluorescence (NRF) measure-ments to verify the authenticity of a warhead. To protect sensitive information, the NRF signal from the warhead is convolved with that of an encryption foil that contains key warhead isotopes in amounts unknown to the inspector. The convolved spectrum from a candidate warhead is then statistically compared against that from an authenticated tem-plate warhead to determine whether the candidate itself is authentic. Since only the final, convolved spectra are observable, and the detailed foil construction is unknown to the in-spector, sensitive information about the warhead is encrypted by physics rather than by software or electronics.

In this thesis, we performed proof-of-concept NRF warhead verification experiments at the High Voltage Research Laboratory (HVRL) at MIT [2]. Using high-purity germa-nium (HPGe) detectors, we measured NRF spectra produced by the interrogation of proxy ‘genuine’ and ‘hoax’ objects by a 2.52 MeV endpoint bremsstrahlung beam. The observed differences in NRF intensities near 2.2 MeV indicate that the physical cryptographic protocol can distinguish between proxy genuine and hoax objects with high confidence. Extrapola-tions to thicker warheads and dedicated verification facilities indicate that realistic warhead verification measurements could be made on the order of hours.

In support of these and future NRF experiments, we also improved and benchmarked the G4NRF code for the simulation of NRF in Geant4 [3]. We first constructed a high-accuracy semi-analytical model for the expected NRF count rate in both simple homogeneous and more complex heterogeneous geometries. We then performed Geant4+G4NRF simulations with these geometries, and found agreement in NRF rates predicted by the semi-analytical model and observed in the simulation at a level of ∼1% in simple test cases and ∼3% in the more realistic complex scenarios. These results improve upon the ∼20% level of the initial G4NRF benchmarking study and establish a highly-accurate NRF framework for Geant4.

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Finally, we conducted a G4NRF validation study using the NRF data taken during the warhead verification experiments [4]. Agreement between the absolute NRF count rates observed in the data and predicted by extensive Geant4+G4NRF modeling validate the combined Geant4+G4NRF model to within ∼20% in the 238_{U NRF transitions and 9%}

in 27_{Al, for an average 14% discrepancy across the entire study. Additionally, agreement}

between the model and data in relative NRF rates is found at the level of .5%. Such agreement in both relative and absolute analyses provides good predictive capability for the design and analysis of future NRF experiments using G4NRF, whether for warhead verification or other applications.

Thesis Supervisor: Areg Danagoulian

Title: Assistant Professor of Nuclear Science and Engineering Thesis Reader: R. Scott Kemp

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“Experience as well as common sense indicated that the most reliable method of avoiding self-extinction was not to equip oneself with the means to accomplish it in the first place.”

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Acknowledgements

To Areg, who taught me the value of concision, To Scott, who taught me to ask interesting questions, To Zach, who taught me fear is the mind-killer, To Brian, who saved me countless months,

To Marina, Rachel, Chat, Pete, Dick, and Brandy, who were always willing to help, To Bob, Brent, Len, Patricia, and Jim, who got me started,

To my family, who persevered,

And to Cody, TD, Adam, Tess, Graeme, Ruaridh, Lee, Spencer, and Noah, who kept me sane, I am richer for having known you.

Jayson Vavrek Cambridge, MA, USA May 2019

The author also acknowledges support from the Consortium for Verification Technology un-der Department of Energy National Nuclear Security Administration award number DE-NA0002534, from the Nuclear Regulatory Commission through a Faculty Development Grant, and from the Massachusetts Green High Performance Computing Center.

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List of Figures

1-1 Overview of the NRF warhead verification measurement. As described in the main text, both a genuine “golden copy” warhead and a candidate warhead are measured using the same encryption foil. The measured NRF spectra are then compared to determine whether the candidate is also genuine. Adapted from Ref. [34]. . . 19 2-1 Partial nuclear level diagram for 238_{U, showing possible NRF transitions}

in-volving the 𝐽𝜋 _{= 1}+ _{state at 𝐸}

𝑟 = 2.209 MeV. On the right, the resonant

level energies 𝐸𝑟 are shown, while on the left, the spin and parity 𝐽𝜋 and level

width Γ𝑟 or lifetime 𝜏𝑟= ~/Γ𝑟 are shown. Emitted photon energies shown do

not account for recoil. . . 22 2-2 The coordinate system used throughout most of this work. The incident

photon beam axis is defined to be the +^𝑧 direction. In the context of Doppler broadening calculations, the target nucleus (shown as a sphere) will have some velocity component 𝑣𝑧 that induces a Doppler shift in energy. In the

context of angular correlation calculations, the vector ⃗𝑣 will instead represent the momentum of the outgoing photon at angle 𝜃 to the incident photon. . . . 23 2-3 NRF cross sections for the 238_{U resonance at 𝐸}

𝑟 = 2.176 MeV. The grey

curve gives the Breit-Wigner cross section at 𝑇 = 0 K, while the blue, yellow, and green curves give the Doppler-broadened cross section at or near room temperature. In the latter cases, the cross section profiles are dominated by Doppler broadening roughly within the thermal width Δ of the resonance (i.e., |𝐸 − 𝐸𝑟| ≃ 3eV) and appear Gaussian; outside of this region, the curves

regain their Lorentzian Breit-Wigner form. . . 25 2-4 Angular correlation functions 𝑊 (𝜃, 𝜑) vs 𝜃 for unpolarized beams. Top:

𝑊 (𝜃, 𝜑)for three transitions in238U. Bottom: 𝑊 (𝜃, 𝜑) for three transitions in

27_{Al. In both plots, the horizontal gridline at 𝑊 (𝜃, 𝜑) = 1 denotes isotropic}

scattering, while the vertical gridline denotes the angle of 𝜃𝑑= 125∘ used in

the experiments of Chapter 4. . . 29 2-5 One-dimensional, single-isotope schematic for the NRF radiation transport

model used to derive the expected NRF count rate. In a warhead verification scenario, the object would be a (multi-isotope) warhead to be measured, and the foil would be a (multi-isotope) encryption foil whose detailed construction (i.e., multiple layers of thicknesses 𝑋𝑘) would be kept secret from the inspector. 30

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2-6 Sketches of the flux energy spectrum throughout the NRF measurement— see Fig. 2-5. Axes not to scale. Left: the incident bremsstrahlung flux, 𝜑0(𝐸), extending from zero to the electron beam energy 𝐸max. Center: the

transmitted flux, 𝜑𝑡(𝐸), with notches at the resonance energies 𝐸𝑟 marked

by dashed lines. Right: the detected flux, 𝑛(𝐸′₎_{. NRF peaks appear at the}

resonance energies and at energies corresponding to branched decays through an intermediate state. In reality, the notches are only Δ ∼ 1 eV wide while the peak widths are ∼1 keV primarily due to detector resolution. . . 30 3-1 Count rates relative to the full numerical integration under approximations

of the NRF cross section, as a function of foil areal density 𝜌𝑋. The four curves correspond to the Gaussian approximation (solid curves) for the27_Al

2.212 MeV and238_{U 2.209 MeV cross sections and their rectangular analogues}

(dashed curves), each with 𝐷 = 0. The rectangular approximation results in up to 20% error for large foil areal densities 𝜌𝑋, while the Gaussian approx-imation results in .2% error. . . 38 3-2 Count rates relative to the full numerical integration under approximations of

the NRF cross section, as a function of measurement object areal density 𝜌𝐷. The four curves correspond to the Gaussian approximation (solid curves) for the27_{Al 2.212 MeV and}238_{U 2.209 MeV cross sections and their rectangular}

analogues (dashed curves), each in the limit 𝑋 → 0. The accuracy of the rect-angular approximation is poor even for modest values of 𝜌𝐷, and eventually reaches a value of zero. The Gaussian approximation retains good accuracy over a larger range of 𝜌𝐷, but eventually decays for very thick targets. . . 39 3-3 Benchmarking of G4NRF with a varying 238_{U areal density 𝜌𝑋. Solid lines}

correspond to the semi-analytical model predictions (via integration of Eq. 2.42), while points correspond to the count rates produced using G4NRF. Here (and in subsequent Figures) error bars denote ±1 𝜎 statistical uncertainties. Error bars are smaller than the size of the markers. . . 43 3-4 Benchmarking of G4NRF with a varying 238_{U areal density 𝜌𝑋 and thin}

(𝐷 = 1 mm) pure 238_{U object. . . 43}

3-5 Benchmarking of G4NRF with a varying 238_{U areal density 𝜌𝑋 and thick}

(𝐷 = 1 cm) pure 238_{U object. Note that unlike in Figs. 3-3 and 3-5, the}

2.245 MeV line is the strongest despite having the weakest cross section. In this thick-target geometry, however, the increase in NRF production rate as the foil thickness 𝑋 increases is overcome by the increase in NRF attenuation in the measurement object as its thickness 𝐷 increases. . . 44 3-6 Benchmarking of G4NRF with a varying 27_{Al areal density 𝜌𝑋. . . 44}

3-7 Benchmarking of G4NRF with a varying 27_{Al areal density 𝜌𝑋 and thin}

(𝐷 = 1 cm) pure27_{Al target. . . 45}

3-8 Benchmarking of G4NRF with a varying 27_{Al areal density 𝜌𝑋 and thick}

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3-9 Benchmarking of G4NRF in the simulations of Ref. [1]. Circles denote the notch-refill-subtracted results and stars denote the raw results. Following the convention of Ref. [1], values of NRF counts 𝑛 are given per 8 × 1011

(importance-sampled) primary bremsstrahlung photons. Statistical uncer-tainties in the notch-refill-subtracted results are smaller than the size of the points. Uncertainties in the unsubtracted results are larger due to the im-portance sampling scheme, since the off-resonant parts of 𝜑0—and thus the

photons that contribute to notch refill—are undersampled compared to the resonant energy bins (see Sections 3.2.2.1, 3.2.2.2, and 3.3.2). . . 46 4-1 Schematic of the experimental setup at HVRL, adapted from Ref. [2].

Detec-tor 2 is the solitary detecDetec-tor at the bottom of the diagram, and DetecDetec-tors 0 and 3 are at the top, with Detector 0 out of the page with respect to 3. As an information security measure, the large Pb shields prevent the HPGe detectors from directly observing the proxy warhead. . . 51 4-2 Geant4 model of the HVRL geometry (to scale). The measurement object

(here plastic and DU) is shown on the left in brown and blue, while the DU and Al foil components are shown on the right in dark and light blue, respectively. The LaBr3detector is not shown. Some shielding elements (including the 2.54

cm Pb filters between the detectors and the foil) have been made translucent in order to show the detector cylinders and measurement object. The volume in the detector housing above Detector 2 is an additional Pb brick added to reduce scatter from above. Image: [62]. . . 52 4-3 Photos of the experimental setup at HVRL (compare to Figs. 4-1 and 4-2).

Left: annotated photo looking upstream. Right: zoomed-in photo showing the collimator exit and template I. The cylinder taped to the lead collimator is an ionization chamber used by the facility operator to monitor the total beam flux. . . 53 4-4 Measured spectra for DU template II (black points) and Pb hoax IIc (red

points). In this and subsequent Figures, error bars are ±1 standard deviation and are statistical uncertainties only. . . 56 4-5 Measured spectra for DU template II, zoomed to show the NRF signal

re-gion. For clarity, the spectrum of hoax IIc is not shown. Arrows indicate the branching relationships from the three main 238_{U lines to the peaks 45 keV}

lower, as well as the non-branching 2.146 MeV238_{U and 2.212 MeV}27_{Al peaks. 58}

4-6 26-parameter Gaussian peak plus exponential background fits to the summed spectra of template II (black points and curve) and hoax IIc (red points and curve). A comparison of spectra for all verification measurements is shown in Table 4.2. . . 59 4-7 Summed spectra (points) and fits (curves) in the template I (black) vs hoax Ia

(red) verification measurement. . . 59 4-8 Summed spectra (points) and fits (curves) in the template I (black) vs genuine

candidate Ig (red) verification measurement. . . 60 4-9 Summed spectra (points) and fits (curves) in the template I (black) vs hoax Ib

(red) verification measurement. . . 60 4-10 Summed spectra (points) and fits (curves) in the template II (black) vs

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4-11 Black: spectrum from template I, detector 3, normalized by live time in-stead of live charge. The y-axis is zoomed in on the 238_{U NRF peaks and}

thus truncates the 2.212 MeV 27_{Al peak. Blue: 24-hr laboratory background}

measurement, normalized by live time. . . 62 4-12 Transmitted spectrum of the bremsstrahlung beam at the encryption foil for

the template I mock warhead and the Black Sea model warhead [1, 14]. Im-age: [62]. . . 65 5-1 A sample, observed, unnormalized NRF spectrum in Detector 3 from

tem-plate I. The corresponding fit (Eq. 4.4, 𝜒2_{/dof = 0.96) is overlain. . . 68}

5-2 Interpolation of the238_{U 2.176 MeV NRF rate vs Γ}

𝑟 table for Detector 0 and

Template II. The horizontal dashed line indicates the observed NRF rate while the vertical dashed line indicates the expected width value of Γ𝑟 = 54.7 meV. 78

5-3 Observed (black) and simulated (blue) photon energy spectra in the LaBr3

detector in the thin genuine (template I) runs (09/13a+09/14b). The simula-tion includes scattering due to NRF and smearing due to detector resolusimula-tion. Image: [62]. . . 81

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List of Tables

3.1 Assumed NRF cross section parameters used in this work. The integrated cross section ∫︀ 𝜎NRF

𝑟 (𝐸) 𝑑𝐸 is directly proportional to the product Γ𝑟𝑏𝑟,0 ≡

Γ𝑟,0 as shown in Eq. 2.16. . . 40

3.2 Minimum, maximum, and root-mean-square (RMS) discrepancies between Eq. 2.42 (using Eq. 2.46 and Eq. 2.48 for the multi-isotope cases) and G4NRF simulations for various benchmarking scenarios. In the last row, the ‘geomet-ric hoax’ object is described in the text and in Ref. [1], and only the notch refill-subtracted results are compared. . . 47 4.1 Compositions of the encryption foil and measurement objects. In the template

and hoax objects, the DU and Pb plates are located between two additional 19 mm high-density plastic layers. The total areal density is computed using the plastic and metal plate thickness(es), excluding the Al foil used to wrap the DU plates for safe handling. . . 51 4.2 Proxy warhead verification measurements . . . 57 4.3 Warhead geometries and approximate detector live charges required for Pb

replacement hoax detection at 5 𝜎. . . 63 5.1 Efficiencies 𝜖geomand 𝜖intused in Eq. 2.47, and fit parameters 𝛽0and 𝛽1in the

𝑧-dependent total detection probability curves of Eq. 5.2. Losses due to dead layers are captured by the values of 𝜖int and 𝛽0. Monte Carlo uncertainties

for these quantities are discussed in Section 5.2.6. . . 72 5.2 Ratio of predicted NRF rates to observed NRF rates for the four models,

averaged over the three detectors and all runs. In Tables 5.2 and 5.3–5.6, the uncertainties in the first three columns derive only from the uncertainty in the 𝑎𝑘 parameter in the fit to Eq. 4.4 (which ultimately arises from counting

statistics in the NRF data and goodness-of-fit), while the last two columns include statistical uncertainty from the brute-force Model III. . . 74 5.3 Ratio of predicted NRF rates to observed NRF rates for the three detectors

and four models, averaged over the three NRF lines and all runs. . . 74 5.4 Ratio of predicted NRF rates to observed NRF rates for the five measurement

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5.5 Ratio of predicted NRF rates to observed NRF rates for the seven measure-ment runs and four models, averaged over the three detectors and three NRF lines. Following the order of Table 5.4, Template I was measured during run 09/13a, and the genuine candidate Ig during run 09/14b. Hoax Ia was measured during run 09/13b, and hoax Ib during run 09/14a. Template II was measured during run 09/15a, hoax IIc during run 09/15d, and hoax IId during run 09/15c. . . 74 5.6 Predicted-to-observed ratios 𝑅 of ratios (i.e., relative validation results) of

NRF lines (Eq. 5.3) within the same spectrum. . . 75 5.7 Summary of level widths Γ𝑟 and ground-state branching ratios 𝑏𝑟,0 from this

work and the literature. In the first column, the relevant table number within the work is given in parentheses. . . 79 5.8 Ratios of simulated to observed count rates in the downstream LaBr3 detector

for 𝐸 > 1.1 MeV across all run dates. Statistical uncertainty in the LaBr3

ratio is negligible. The HPGe ratios from the final column of Table 5.5 are included for comparison; the two dates with the most extreme values in the LaBr3dataset (09/13b and 09/15a) correspond to the two most extreme dates

in the HPGe dataset. This extreme behavior is reduced somewhat when normalizing the HPGe ratios against their corresponding LaBr3 ratios. The

LaBr3-normalized NRF data has a study-wide average model-to-data ratio of

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Chapter 1

Introduction

Despite the end of the Cold War, the United States and Russia still maintain thousands of nuclear weapons. Both countries have agreed, at least in principle, to eliminate their arsenals under the Treaty on the Non-Proliferation of Nuclear Weapons (NPT), and have made gradual progress towards this goal. Several landmark treaties such as SALT I and II, SORT, INF, and finally New START have been instrumental in bringing arsenal numbers down to approximately 1700 deployed warheads per side [5, 6].

Even given these reductions, the threat to humanity posed by these remaining nuclear weapons should not be understated. A single modern weapon detonated in the heart of Manhattan or New Delhi could kill over one million civilians.1 _{A handful of weapons on ten}

cities would be “a disaster beyond history” [8]. And a full-scale nuclear war among ther-monuclear powers would be a global catastrophic risk [9, 10] critically threatening modern civilization.

The likely expiration of New START in 2021—the only remaining treaty that places limits on US and Russian nuclear arsenals—is therefore of great concern to the arms control community. Fears of a new arms race unchecked by numerical limits have spurred calls for the extension of New START and, looking forward, the negotiation of a new treaty enabling even deeper cuts to US and Russian arsenals. If deeper cuts are to be reached, however, it is likely that parties to the treaty will require some mechanism to directly verify the dismantlement of nuclear warheads and thus the compliance with stricter numerical limits. Any such mechanism proposed to verify the authenticity of nuclear warheads faces two primary challenges. First, there has been substantial sociopolitical opposition to the end goal of eliminating what are seen as effective nuclear deterrents and thereby risking the ostensible ‘nuclear peace’ that has existed since 1945 [11, 12]. The second challenge, and the one on which this thesis focuses, is not sociopolitical but technological: nations will not agree to any verification measurement that reveals sensitive information about the design of their warheads, but a measurement that is sufficiently useful for verification is likely highly intrusive [13, 14]. The development of an information-secure warhead verification protocol is therefore a probable future bottleneck to further successful nuclear arms reductions, and should be investigated well in advance of any negotiations that aim to implement a direct nuclear warhead verification regime.

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1.1 Warhead verification

In a warhead verification protocol, a warhead owner (‘host’) attempts to prove to an inspec-tion team (‘inspector’) that an object submitted for inspecinspec-tion and subsequent dismantle-ment and disposition is indeed a genuine nuclear warhead. An object successfully verified may then be dismantled by the host under a secure chain of custody [15] and counted to-wards the host’s obligations under an arms reduction treaty. At the same time, the host seeks to prevent the inspector from learning any sensitive information about the design of the warhead, whether to prevent proliferation of nuclear weapons technology or disclosure of warhead architecture and vulnerabilities. Thus, the verification measurement must be designed and performed in such a way as to provide a strong test of authenticity while while minimizing intrusiveness and maximizing information security. Non-authentic war-heads (‘hoaxes’) fall into two broad categories: isotopic hoaxes, in which a valuable weapon component (e.g., the weapons-grade Pu fissile fuel) is replaced by a less-valuable surrogate of similar geometry (e.g., reactor grade Pu); and geometric hoaxes, in which isotopes are present in their correct amounts but in a non-weapons-usable configuration (e.g., rough slabs of Pu rather than highly-engineered spherical shells).

Past approaches to warhead verification2 _{have generally focused on the ‘attribute’}

ap-proach, in which the protocol measures a set of key characteristics thought to define a warhead, such as the total mass of plutonium and the isotopic ratio of 239_{Pu to} 240_{Pu in}

the object [16]. Such measurements are highly intrusive, and so are conducted behind an ‘information barrier’ (IB), an electronic or software layer that shields the classified raw mea-surement data and presents the inspector with only a binary pass/fail answer for each of the attribute measurements [17]. However, certifying that an electronic or (especially) software IB does not contain any hidden backdoors or functionalities—which a nefarious inspector could exploit to obtain sensitive information or a nefarious host could use to fraudulently simulate a ‘genuine’ result—is exceedingly difficult, and may never be satisfactorily proven. Moreover, attributes must be chosen specifically to describe real nuclear warheads, and thus may constitute sensitive information themselves. Even then, the set of attributes may not be complete, opening the door to hoax objects that pass all the attribute tests but nevertheless are not real warheads.

More recent work has therefore focused on the ‘template’ approach to verification, in which comparison to a known genuine object (the “golden copy”) is used to authenticate subsequent objects presented for inspection [18, 19, 20]. In such a protocol, the measure-ments of both the golden copy and subsequent candidate objects are encrypted using the same method, so that only the encrypted signals (or “hashes”) must be compared to authen-ticate. The hash should be unique to a particular combination of geometry and isotopic makeup (i.e., a particular warhead design), while containing no sensitive information about the object. As such, the hash is useless on its own, and only has any use in comparison against the hash of another object—a warhead that is already known to be genuine. This authenticated golden copy warhead could be established for instance via an unannounced visit by the inspector to a random launch facility in the host country, and then by selecting a random warhead from an active-duty intercontinental ballistic missile.3 _{A measurement of} 2_{Treaties such as New START have bypassed the difficult issue of warhead verification and instead}

focused on the verification of warhead delivery vehicles such as bomber aircraft and intercontinental ballistic missiles (ICBMs).

3_{In any template warhead verification protocol, the utility of every measurement hinges on the}

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the authenticated golden copy (called the “template measurement”) would then be used as the standard against which to compare the measurements from the same model of warhead covered by the arms control treaty.4

Recent papers have put forth template verification protocols that aim to make a verifi-cation measurement of a warhead while protecting sensitive design information. A team of researchers at Princeton proposed and later experimentally demonstrated a verification pro-tocol using superheated bubble detectors and fast neutron radiography [21, 22]. In parallel, a team (including the author) at the Massachusetts Institute of Technology (MIT) developed an alternative approach using isotopic tomography via transmission nuclear resonance fluo-rescence (NRF) [1, 2]; the present work contains an experimental demonstration of the MIT NRF protocol. Further techniques using coded-aperture-based passive neutron counters [23] and epithermal neutron resonance radiography [24], from Sandia National Laboratories and MIT, respectively, have been proposed in the past few years.

The strengths and weaknesses of the aforementioned proposals can be compared by examining the three requirements of an ideal warhead verification protocol:

1. completeness: the ideal protocol must clear all real warheads;

2. soundness: the ideal protocol must raise an alarm on all hoax warheads;

3. information security: the ideal protocol must be zero-knowledge [25, 26]—for an honest host, it must not reveal anything beyond a binary genuine/hoax determination. The Princeton protocol is essentially zero-knowledge, returning a flat image (up to statistical variation) if the host has submitted a real warhead. In its original form [21], the measure-ment faces a challenge in the soundness requiremeasure-ment: fast neutron radiography is insensitive to the isotopic or (in some cases) elemental composition of the object, and cannot on its own distinguish between weapon materials and well-chosen hoax materials. Additional measure-ment modes using multiple incident neutron energies [27] have been proposed to increase the protocol’s discrimination between fissionable and fissile isotopes. Similarly, work on the Sandia coded-aperture protocol has focused on satisfying the completeness and information security aspects of the problem, but has not demonstrated resistance to hoaxing by a neutron source of similar geometry and activity.

Unlike the Princeton and Sandia protocols, the two MIT protocols are highly sensitive to isotopics through their use of isotope-specific resonant phenomena, making them highly robust against a large class of hoaxes. While the MIT NRF protocol is not zero-knowledge (since the inspector has access to the hashed measurements rather than solely a binary genuine/hoax determination), and thus there are uncertainties about the extent of the in-formation security of the MIT NRF protocol, there are methods to make it sufficiently secure [1]. This work demonstrates the core measurement of the MIT NRF protocol in Chapter 4, an experimental implementation of an isotopically-sensitive warhead verification measurement.

will require classified knowledge of the chain of custody of a country’s nuclear stockpile, and therefore is an open question beyond the scope of this thesis.

4_{The term ‘template’ is used for the measurement data from the golden copy, but also less formally (e.g.,}

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1.2 Nuclear resonance fluorescence

Nuclear resonance fluorescence (NRF) describes the X(𝛾, 𝛾)X or X(𝛾, 𝛾′₎_{X reaction in which}

a photon 𝛾 is resonantly absorbed by the nucleus X and then re-emitted as the excited nucleus subsequently transitions to its ground state [28, 29]. Fundamentally, this interaction arises from resonant solutions of the Schrödinger equation, and as such, has an interaction cross section with a width inversely proportional to the lifetime of the excited state. These widths are typically ∼10 meV for high-𝑍 isotopes of interest, but the effective width of the NRF cross section is increased to ∼1 eV through Doppler broadening by thermal motion of the target nuclei. Imperfect detector resolution further broadens the measurable NRF resolution to widths of ∼1 keV. Since the NRF lines of an isotope are still typically >10 keV apart, the set of resonance energies 𝐸𝑟 provides a resolvable, one-to-one map between measurement space

and isotopic space that has been used in a variety of non-destructive assay applications [30, 31, 32, 33].

The MIT verification protocol exploits the isotope-specific nature of NRF to make a template measurement of the mass and geometry of the isotopes of interest to the inspector. Such isotopes would likely include U and/or Pu isotopes in the fissile warhead components; N, C, and O isotopes in the conventional high explosives; and possibly isotopes of Fe and Al in the structural components of the weapon. As discussed in Chapters 2 and 4, the measurement uses a broad-spectrum bremsstrahlung photon source to irradiate the warhead; NRF interactions in the warhead preferentially attenuate the photon flux at specific energies determined by the unique nuclear energy level structure of each isotope according to the amount of the isotope present in the warhead.5 _{The remaining transmitted flux at these}

energies goes on to induce further NRF interactions in an encryption foil, leading to NRF emission into high-purity germanium (HPGe) photon detectors at an observed rate (the later Eq. 2.42) that has been reduced by the presence of the NRF isotope in the warhead— see Fig. 1-1. The hashed measurements required for the template verification protocol are thus the recorded spectra, since it is impossible to precisely determine the warhead composition (i.e., the thickness 𝐷 in the later Eqs. 2.34 and 2.42) from the height of the NRF peaks in the observed spectrum without knowledge of the detailed composition of the foil (i.e., the thickness 𝑋 in the later Eq. 2.42). The exact foil design is therefore decided by the host and kept secret from the inspector. The influence of the warhead composition on the height of the NRF peaks—and thus any sensitive warhead design information—is then said to be physically encrypted by the foil. This technique uses the laws of physics to mask sensitive information, rather than electronic or computer-based information barriers, making it substantially more robust against tampering and hoaxing than previously proposed techniques [17].

Although the detailed construction of the foil is kept secret from the inspector in order to maintain the encryption, the mere presence of certain characteristic NRF lines in the detected spectrum corresponds to the presence of certain isotopes in the encryption foil, a fact the inspector may use to validate the utility of a given foil without breaking the encryption. The foil may also be placed under joint custody of the host and inspector to ensure it has not been altered between the template and candidate measurements. As an additional layer of information security, the host may add optional ‘encryption plates’ 5_{The analogous process of atomic fluorescence, in which eV-scale photons resonantly trigger excited}

electronic states, is responsible for phenomena such as element-specific absorption lines in the blackbody spectra of stars.

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2-3 MeV electrons

HPGe

bremsstrahlung photon spectrum

collimator

shield

candidate warhead

NRF-notched transmitted spectrum

candidate NRF spectrum encryption foil

?

genuine warhead genuine NRF spectrum =?

Figure 1-1: Overview of the NRF warhead verification measurement. As described in the main text, both a genuine “golden copy” warhead and a candidate warhead are measured using the same encryption foil. The measured NRF spectra are then compared to determine whether the candidate is also genuine. Adapted from Ref. [34].

of warhead materials to the measured object so that even if precise inference about the measured object is possible, it is impossible to infer anything about the warhead alone.

To protect against geometric hoaxes, the NRF protocol includes measurements of tem-plate and candidate warheads in random or multiple random orientations due to the difficulty for the host to engineer a hoax warhead that could mimic the template signal successfully along multiple randomly-selected projections. To increase the information security of this protocol, each orientation may be paired with a unique cryptographic foil to dilute the in-formation content of the multiple measurements. Ref. [1] discusses the required complexity of such geometric hoaxes, which increases rapidly with number of projections measured.

1.3 Structure and goals

The primary goal of this thesis is to cover the development of an NRF-based isotope-sensitive warhead verification technique, starting from the rather fundamental physics of NRF inter-actions and culminating in an experimental demonstration of the NRF measurement. A secondary goal is to describe the benchmarking and validation of the G4NRF code that enabled the design and analysis of the experimental demonstration, and will be useful in future NRF studies. As such, this thesis has been synthesized from four of the author’s manuscripts—Refs. [1, 2, 3, 4]—which have either been published or are in production at

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the time of writing.6,7

Chapter 1, the current introductory chapter, describes the NRF protocol first devel-oped in Kemp, Danagoulian, Macdonald, and Vavrek, PNAS 2016, Physical cryptographic verification of nuclear warheads[1]. This chapter introduces the problem of nuclear warhead verification, gives an overview of NRF, and motivates the development of the NRF protocol beyond the Monte Carlo simulations of Ref. [1] (and the author’s SM thesis [35]).

Chapter 2provides the theory of NRF physics following the treatment in Vavrek et al., NIM B 2018, High-accuracy Geant4 simulation and semi-analytical modeling of nuclear res-onance fluorescence [3], and references therein. It then constructs a high-accuracy semi-analytical model for NRF count rates in a typical NRF measurement. Some changes in notation from the paper have been made for better internal consistency.

Chapter 3 is also based on Ref. [3], and contains a series of NRF simulations bench-marked against the semi-analytical model developed in Chapter 2.

Chapter 4discusses the main result of this work, an experimental demonstration of the NRF warhead verification measurement, as detailed in Vavrek et al., PNAS 2018, Exper-imental demonstration of an isotope-sensitive warhead verification technique using nuclear resonance fluorescence [2].

Chapter 5compares the NRF data taken in Chapter 4 against four absolute NRF count rate models based on G4NRF and/or the semi-analytical model of Chapter 2. This analysis has been presented in a preprint, Vavrek et al., Absolute validation of nuclear resonance fluorescence models and cross sections in 238U and 27Al (working title) [4] and is currently in preparation for submission as a journal publication.

Chapter 6 concludes with a discussion of future research directions. Possible vulner-abilities of the NRF protocol—some of which have proposed solutions, and some of which require further investigation—are listed, as are broader research directions aimed at ad-vancing the protocol beyond an academic proof-of-concept. Future research directions for G4NRF are also covered.

Now, with preliminaries out of the way, we can turn our attention to the underlying physics of NRF, and build our way up from the Schrödinger equation to the secure disar-mament of nuclear warheads.

6_{PNAS automatically grants authors permission to include their own articles as part of their dissertations:}

https://www.pnas.org/page/about/rights-permissions

7_{Elsevier automatically grants authors permission to include their own articles, in full or in part, in a}

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Chapter 2

Theory of nuclear resonance

fluorescence

In this chapter, we develop a high-accuracy model of the microscopic NRF cross section and macroscopic NRF radiation transport behavior. Such a detailed theoretical model will prove useful for subsequent chapters, where it will be used to help design the experiments of Chapter 4 and to benchmark and validate the Geant4+G4NRF simulation framework in Chapters 3 and 5.

2.1 Microscopic picture of NRF

Nuclear resonance fluorescence (NRF) is the X(𝛾, 𝛾)X or X(𝛾, 𝛾′₎_{X photonuclear interaction}

in which a nucleus X with resonant energy level 𝐸𝑟 absorbs a photon 𝛾 of energy 𝐸 ≃ 𝐸𝑟,

causing the nucleus to transition from its ground state to the excited state8 _{with energy 𝐸} 𝑟.

These electromagnetic transitions between states are typically the M1, E1, or E2 modes, and the 238_{U excitations near 2.2 MeV that are a focus of this work are described by the}

M1 ‘scissors modes’ in which either groups of neutrons and protons (‘orbital scissors’) or spin-up and spin-down nucleons (‘spin scissors’) oscillate against each other [29, 36, 37].

The excited nuclear state typically has a width of Γ𝑟 ∼ 1–100 meV and corresponding

𝑒-folding lifetime of 𝜏𝑟 = ~/Γ𝑟∼ 10–1000 fs. On this timescale, the excited state decays either

to a low-lying intermediate state with energy 𝐸𝑗 or directly to the ground state (𝐸𝑗 ≡ 0) by

emitting a photon9 _{of energy 𝐸}′ _{= 𝐸 − 𝐸}

𝑗 (see Fig. 2-1).

2.1.1 NRF cross sections

At temperature 𝑇 = 0 K, the microscopic NRF cross section for absorption through an isolated resonant energy level 𝐸𝑟 followed by decay to level 𝐸𝑗 is the familiar Breit-Wigner 8_{This introductory description neglects the small kinetic energy of the nuclear recoil required for}

mo-mentum conservation. Due to the non-zero recoil, the photon energy required to trigger an NRF interaction is slightly larger than the level energy. This section does not draw a distinction between the level energy and the resonance energy, but the topic will be taken up again in Section 2.1.2.

9_{In some cases is also possible for the excited state to decay through other processes, such as internal}

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0 0+ 2.245 MeV 1+, 29 meV 2.209 MeV 1+, 54 meV 2.176 MeV 1+, 55 meV 0.045 MeV 2+, 206 ps 𝐸 ≃ 𝐸𝑟 𝐸′= 𝐸 𝐸′ = 𝐸 − 𝐸𝑗 238

_U

Figure 2-1: Partial nuclear level diagram for238_{U, showing possible NRF transitions}

involv-ing the 𝐽𝜋 _{= 1}+ _{state at 𝐸}

𝑟= 2.209 MeV. On the right, the resonant level energies 𝐸𝑟 are

shown, while on the left, the spin and parity 𝐽𝜋 _{and level width Γ}

𝑟 or lifetime 𝜏𝑟 = ~/Γ𝑟

are shown. Emitted photon energies shown do not account for recoil. cross section [28], 𝜎_{𝑟,𝑗; 0 K}NRF (𝐸) = 𝜋𝑔𝑟 (︂ ~𝑐 𝐸 )︂2 Γ𝑟,0Γ𝑟,𝑗 (𝐸 − 𝐸𝑟)2+ (Γ𝑟/2)2 , (2.1)

where Γ𝑟,𝑗 denotes the partial width for transitions between states 𝑟 and 𝑗,

Γ𝑟≡

∑︁

𝑗

Γ𝑟,𝑗 (2.2)

is the total width of the level, and

𝑔𝑟≡

2𝐽𝑟+ 1

2(2𝐽0+ 1) (2.3)

is the spin statistical factor10 _{determined by the resonant and ground state spins 𝐽} 𝑟 and

𝐽0. Eq. 2.1 itself may be derived using perturbation theory [38] or, as promised in

Chap-ter 1, from the time-independent Schrödinger equation via the 𝑅-matrix method [39]—see Appendix A.

10_{The inclusion of the factor of 2 in the denominator of 𝑔}

𝑟(rather than directly in Eq. 2.1) is a convention

that differs throughout the literature. We retain the 2 in 𝑔𝑟 as it arises from the two photon polarization

states replacing the number of spin states for a massive incident particle in the traditional spin statistical factor [38].

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+^

𝑥

+^

𝑦

+^

𝑧

⃗

𝑣

𝑧

𝜑

𝜃

𝐸

Figure 2-2: The coordinate system used throughout most of this work. The incident photon beam axis is defined to be the +^𝑧 direction. In the context of Doppler broadening calcu-lations, the target nucleus (shown as a sphere) will have some velocity component 𝑣𝑧 that

induces a Doppler shift in energy. In the context of angular correlation calculations, the vector ⃗𝑣 will instead represent the momentum of the outgoing photon at angle 𝜃 to the incident photon.

When developing the macroscopic picture of an NRF measurement in Section 2.2, we will often be concerned with the unattenuated (i.e., non-interacting) flux. Thus it is often more convenient to work with the NRF absorption cross section, i.e., Eq. 2.1 summed over all decay modes:

𝜎_{𝑟; 0 K}NRF(𝐸) =∑︁ 𝑗 𝜎_{𝑟,𝑗; 0 K}NRF (𝐸) = 𝜋𝑔𝑟 (︂ ~𝑐 𝐸 )︂2 Γ𝑟,0Γ𝑟 (𝐸 − 𝐸𝑟)2+ (Γ𝑟/2)2 . (2.4)

The absorption+decay cross section of Eq. 2.1 differs from the absorption-only cross section of Eq. 2.4 by the ratio of widths, also known as the branching ratio

𝑏𝑟,𝑗 ≡

Γ𝑟,𝑗

Γ𝑟

. (2.5)

At non-zero temperatures, the target nuclei will have some non-zero distribution of velocities due to thermal motion; velocity components parallel or anti-parallel to the incident photon momentum will induce Doppler red- or blue-shifts, respectively. A common model for the target nuclei velocity distribution is a modified Maxwell-Boltzmann distribution using the Lamb temperature correction to account for the non-ideal-gas lattice structure of atoms in a solid [28]. The distribution of velocity components 𝑣𝑧 along a 𝑧-axis chosen to align

with the incident photon momentum is given by 𝑤(𝑣𝑧)𝑑𝑣𝑧 = √︂ 𝑀 2𝜋𝑘𝐵𝑇 exp (︂ −𝑀 𝑣 2 𝑧 2𝑘𝐵𝑇 )︂ 𝑑𝑣𝑧, (2.6)

where 𝑀 is the target nucleus mass, 𝑇 is the absolute temperature of the target material, and 𝑘𝐵is Boltzmann’s constant. According to the convention of Fig. 2-2, 𝑣𝑧 < 0corresponds

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to the nucleus and photon moving towards each other, in which case the Doppler-shifted energy ^𝐸 is given by ^ 𝐸 = 𝐸 √︃ 1 − 𝑣𝑧/𝑐 1 + 𝑣𝑧/𝑐 ≃ 𝐸(︁1 −𝑣𝑧 𝑐 )︁ , (2.7)

where the non-relativistic approximation 𝑣𝑧/𝑐 ≪ 1 has been applied, giving

𝑤( ^𝐸)𝑑 ^𝐸 = √1 𝜋Δexp [︃ −( ^𝐸 − 𝐸) 2 Δ2 ]︃ 𝑑 ^𝐸, (2.8)

where we have defined the Doppler width as Δ ≡ 𝐸

√︂ 2𝑘𝐵𝑇

𝑀 𝑐2 . (2.9)

Averaging Eq. 2.4 over 𝑤( ^𝐸), we obtain the Doppler-broadened Breit-Wigner NRF absorp-tion cross secabsorp-tion,

𝜎_𝑟NRF(𝐸) = 2𝜋1/2𝑔𝑟 (︂ ~𝑐 𝐸𝑟 )︂2 𝑏_√𝑟,0 𝑡 ∫︁ +∞ −∞ exp [︂ −(𝑥 − 𝑦) 2 4𝑡 ]︂ 𝑑𝑦 1 + 𝑦2, (2.10)

where we have replaced 𝐸 → 𝐸𝑟 in the denominator (valid near the resonance) and used

the standard definitions

𝑥 ≡ 2(𝐸 − 𝐸𝑟)/Γ𝑟, (2.11)

𝑦 ≡ 2( ^𝐸 − 𝐸𝑟)/Γ𝑟, (2.12)

𝑡 ≡ (Δ/Γ𝑟)2. (2.13)

The aforementioned Lamb temperature correction, which accounts for the lattice structure of the target atoms, replaces the true temperature 𝑇 of the system with an effective temperature

𝑇eff= 3𝑇 (︂ 𝑇 𝜃𝐷 )︂3∫︁ 𝜃𝐷/𝑇 0 𝑥3 (︂ 1 𝑒𝑥_{− 1}+ 1 2 )︂ , (2.14)

where 𝜃𝐷 is the Debye temperature of the target material.

In the common case that Δ ≫ Γ𝑟, i.e., 𝑡 ≫ 1, the thermal width dominates the NRF

cross section, and, eliminating 𝑦 term in the exponential, Eq. 2.10 can be written in the Gaussian form 𝜎_𝑟Gaus(𝐸) = 2𝜋3/2𝑔𝑟 (︂ ~𝑐 𝐸𝑟 )︂2 𝑏0,𝑟 Γ𝑟 Δ exp [︂ −(𝐸 − 𝐸𝑟) 2 Δ2 ]︂ (2.15) which has a centroid 𝐸𝑟and variance Δ2/2. The modified integrand in Eq. 2.10 will then be

inaccurate for large values of |𝑦|, although the large-|𝑦| contributions to the integral in the exact Eq. 2.10 are small. Similarly, because the Gaussian tails of Eq. 2.15 fall off much more rapidly than the true Lorentzian tails of Eq. 2.10, the pure Gaussian form will be inaccurate for large |𝑥|—see, e.g., Fig. 2-3. Again, though, the inaccuracies are located in regions of low cross section, and so their influence on predicted NRF interaction rates will often be

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270 K 300 K 330 K 0 K(BW) -4 -2 0 2 4 0.1 1 10 100 1000 E-Er [eV] σ (E ) [b ]

Figure 2-3: NRF cross sections for the238_{U resonance at 𝐸}

𝑟 = 2.176MeV. The grey curve

gives the Breit-Wigner cross section at 𝑇 = 0 K, while the blue, yellow, and green curves give the Doppler-broadened cross section at or near room temperature. In the latter cases, the cross section profiles are dominated by Doppler broadening roughly within the thermal width Δ of the resonance (i.e., |𝐸 −𝐸𝑟| ≃ 3eV) and appear Gaussian; outside of this region,

the curves regain their Lorentzian Breit-Wigner form.

small. This question will be taken up quantitatively starting in Section 3.1.1.

It is common in the literature to define the energy-integrated cross section as a useful measure of the overall ‘strength’ of a resonance:

𝜎_𝑟int≡ ∫︁ 𝜎_𝑟NRF(𝐸) 𝑑𝐸 = 2𝜋2𝑔𝑟 (︂ ~𝑐 𝐸𝑟 )︂2 Γ𝑟,0, (2.16)

assuming 𝐸𝑟 ≫ Δ ≫ Γ𝑟. As we will show in Section 2.2, the predicted count rate is

proportional to 𝜎int

𝑟 in the thin target limit (Eq. 2.50), eliminating any dependence on the

precise shape of the NRF cross section (Eq. 2.10).

An even rougher approximation to Eq. 2.10 is the ‘rectangular cross section,’ i.e., a constant value of cross section over a narrow range of energies 𝐸 [40]. The energy range is arbitrary but typically chosen to be [𝐸𝑟− Δ𝐻/2, 𝐸𝑟+ Δ𝐻/2], where

Δ𝐻 ≡ 4

√

ln 2 Δ (2.17)

is the full-width at half-maximum of the Gaussian profile of Eq. 2.15, and the height is chosen to preserve the integrated cross section of Eq. 2.16 so as to maintain accuracy for thin targets. Specifically,

𝜎_𝑟rect(𝐸) ≡ 1 (|𝐸 − 𝐸𝑟| ≤ Δ𝐻/2) ·

𝜎int_𝑟

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where 1(·) is equal to 1 if its argument is true and 0 otherwise. As will be shown in Section 3.1.1, NRF count rates using the rectangular cross section approximation will be significantly inaccurate for thick targets.

2.1.2 Nuclear recoil

In order to satisfy both conservation of energy and momentum simultaneously, a free nucleus undergoing NRF will recoil from the interaction with a non-zero kinetic energy. The most common recoil model in the literature is a simple two-step model that neglects angular dependencies and thermal motion. In the first step, a nucleus of mass 𝑀 at rest in its ground state absorbs a photon of energy 𝐸 and is promoted to an excited state at 𝐸*_.

Conservation of energy and linear momentum give 𝐸 = 1

2𝑀 𝑣

2_{+ 𝐸}*_, _(2.19)

𝐸

𝑐 = 𝑀 𝑣. (2.20)

Equating the 𝑀𝑣2 _{derived from both equations then gives}

𝐸 − 𝐸* = 𝐸

2

2𝑀 𝑐2, (2.21)

which is the energy lost by the photon, i.e., converted to the kinetic energy of the nucleus, during the absorption step. In the emission step, the excited nucleus is again assumed to be initially at rest for simplicity of calculation, then recoils with speed 𝑣′ _{after emitting a}

photon of energy 𝐸′_{, in which case the conservation equations are}

𝐸*= 1 2𝑀 𝑣 ′2_{+ 𝐸}′ _(2.22) 0 = −𝑀 𝑣′+ 𝐸 ′ 𝑐 . (2.23)

Again equating the 𝑀𝑣′2 _{terms, we find that}

𝐸*− 𝐸′ = 𝐸

′2

2𝑀 𝑐2 (2.24)

is the energy lost by the photon to the nuclear recoil during the emission step. Altogether, the total energy transferred to nuclear recoil in both steps is

𝐸rec ≡ 𝐸 − 𝐸′ =

𝐸2+ 𝐸′2 2𝑀 𝑐2 ≃

𝐸2

𝑀 𝑐2. (2.25)

Another model [34] assumes a single instantaneous scattering interaction with a free nucleus at rest, in which case the kinematics are identical to Compton scattering and the recoil energy is 𝐸rec= 𝐸 [︂ 1 − 1 1 + 𝐸(1 − cos 𝜃)/𝑀 𝑐2 ]︂ ≃ 𝐸 2 𝑀 𝑐2(1 − cos 𝜃), (2.26)

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the approximation 𝐸 ≪ 𝑀𝑐2 _{has been applied. Values of 𝐸}

rec vary from 0 in the forward

(𝜃 = 0) direction to 2𝐸2_{/𝑀 𝑐}2 _{in the backward (𝜃 = 𝜋) direction. For a photon energy}

of 𝐸 = 2.2 MeV incident on 238_{U or} 27_{Al and re-emitted at the 𝜃 = 125}∘ _{commonly used}

in experiments, 𝐸rec ≃ 35 eV or 300 eV, respectively. Over all solid angles (ignoring the

angular correlations of Section 2.1.3), the average value of 𝐸rec is

⟨𝐸rec⟩ = 𝐸2 𝑀 𝑐2 · ∫︀2𝜋 0 ∫︀𝜋 0 (1 − cos 𝜃) sin 𝜃 𝑑𝜃𝑑𝜑 ∫︀2𝜋 0 ∫︀𝜋 0 sin 𝜃 𝑑𝜃𝑑𝜑 = 𝐸 2 𝑀 𝑐2, (2.27)

which matches the two-step, 1D result of Eq. 2.25.

Since the outgoing photon energy is reduced by 𝐸rec≫ Δ, photons emitted in the

back-wards direction (which produce comparatively large nuclear recoils) will be well off-resonance and highly unlikely to undergo another NRF interaction. This lack of self-attenuation in the backwards direction, along with high backgrounds in the forward direction, gives rise to experiments in which NRF photon spectra are measured in backwards emission directions, as discussed in Section 2.2.

The recoil loss is often ignored because 𝐸rec ≪ 𝐸𝑟, leading to some ambiguity among

the resonance energy 𝐸𝑟, the level energy 𝐸*, the incident photon energy 𝐸, and, if the

decay is directly to the ground state, the emitted photon energy 𝐸′_{. For concreteness, we}

define the level energy 𝐸* _{to be the fundamental difference in energies between the ground}

and excited states, while the resonance energy is the energy for which the NRF interaction cross section (Eq. 2.10) is maximized, taking into account the nuclear recoil. For simplicity, the G4NRF code computes the resonance energy as 𝐸𝑟 = 𝐸* + 𝐸2/2𝑀 𝑐2 based on the

absorption step recoil of Eq. 2.21, and then decrements the final-state photon energy 𝐸′ _by

another 𝐸2_{/2𝑀 𝑐}2_.

The two recoil models captured in Eqs. 2.25 and 2.26 neglect several effects, including the thermal motion of the nucleus as discussed in Section 2.1.1, the binding energy of the nucleus in its atomic lattice, and the lifetime of the excited state. For nuclei bound in an atomic lattice, 𝐸rec may be large enough to overcome the lattice displacement energy 𝐸𝑑

(&10 eV in pure metals [41]) in which case the kinetic energy transfer is 𝐸rec−𝐸𝑑. If the value

of 𝐸rec computed for an unbound nucleus is less than 𝐸𝑑, the recoil is instead transferred

to the entire lattice: 𝑀 → ∞, 𝐸rec → 0, and recoilless NRF (the Mössbauer effect) is

achieved [42]. Additionally, the lifetime 𝜏𝑟= ~/Γ𝑟 of the excited state may be shorter than

the time required for the recoiling nucleus to recombine with the atomic lattice. In this case, the excited state can decay in-flight, introducing an additional Doppler shift to the outgoing photon energy 𝐸′_.

2.1.3 Angular dependencies

So far, all the NRF cross section equations have been functions of energy 𝐸 alone. More generally, the NRF differential cross section contains an angular dependence, which we may denote as 𝑑𝜎NRF_𝑟 (𝐸) 𝑑Ω = 𝑊 (𝜃, 𝜑) 4𝜋 𝜎 NRF 𝑟 (𝐸), (2.28)

where 𝑊 (𝜃, 𝜑) is the angular correlation function for emission into the infinitesimal solid angle 𝑑Ω = 𝑑cos 𝜃 𝑑𝜑. For unpolarized incident beams, the 𝜑-dependence disappears, and

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𝑊 (𝜃, 𝜑)takes the form

𝑊 (𝜃, 𝜑) = 1 + 𝑎 cos

2_{𝜃 + 𝑏 cos}4_𝜃

1 + 𝑎/3 + 𝑏/5 , (2.29)

where 𝜃 is the angle between incident and outgoing photon momentum directions (equivalent to the polar angle given in Fig. 2-2) and

∫︁ 2𝜋

0

∫︁ 𝜋

0

𝑊 (𝜃, 𝜑) sin 𝜃𝑑𝜃𝑑𝜑 = 4𝜋. (2.30)

The factors 𝑎 and 𝑏 give the dipole and quadrupole transition contributions, respectively, and are functions (tabulated in Ref. [43]) of the three level spins 𝐽0 → 𝐽𝑟→ 𝐽𝑓 involved in

the NRF interaction. The 238_{U ground state transitions studied in this work, for instance,}

have spin sequences 0 → 1 → 0, and thus 𝑎 = 1 and 𝑏 = 0, giving 𝑊010(𝜃, 𝜑) =

3

4(︀1 + cos

2_{𝜃)︀ ,} _(2.31)

which is very nearly equal to 1 at the detector angle of 𝜃 = 125∘ _{commonly used in}

experiments—see Fig. 2-4. For transitions not involving a spin-0 state, such as the 1 → 2 branched decays of the aforementioned238_{U states, the transition can proceed either through}

dipole or quadrupole modes, with corresponding 𝑊 (𝜃, 𝜑): 𝑊₀₁₂dd(𝜃, 𝜑) = 9 8 (︂ 1 −1 3cos 2_𝜃 )︂ (2.32) 𝑊₀₁₂dq(𝜃, 𝜑) = 7 8 (︂ 1 +3 7cos 2_𝜃 )︂ . (2.33)

The division of decays between the two modes is given by the mixing ratio 𝛿 (when known). The 𝑊 (𝜃, 𝜑) are more complicated still for the 5/2 → 7/2 → 5/2 spin sequence of 27_Al,

which can proceed via dipole-dipole, dipole-quadrupole, dipole, or quadrupole-quadrupole pairs of transitions.

2.2 NRF count rate model

In this section, we will use the microscopic NRF cross section to determine the macroscopic behavior of the photon flux through an NRF measurement such as the one depicted schemat-ically in Fig. 1-1. As developed below, the detected NRF energy spectrum will consist of a series of sharp emission lines at the nuclear resonance energies 𝐸𝑟, on top of a smooth

background (see Section 2.2.3). The peak energies can be used to identify the presence of a particular isotope in a sample, and the peak areas (i.e., the number of counts in the peak) encode information about the concentration of that isotope.

Qualitatively, in the so-called ‘transmission mode’ of Fig. 1-1, the measurement object and a reference foil of known construction are both irradiated by a beam of photons. Some of the photons in the incident beam will happen to be on resonance (see, e.g. Fig. 2-3), and thus capable of undergoing NRF interactions in the measurement object or reference foil. The foil would thus produce a known, baseline emission rate of NRF photons if irradiated with no measurement object in the beam. The measurement object acts in a similar fashion,

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W(θ,ϕ): 0,1,0 W(θ,ϕ): 0,1,2, dd W(θ,ϕ): 0,1,2, dq 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 emission angle θ/π angular correlation W (θ ,ϕ ) W(θ,ϕ): 5/2,7/2,5/2, dd W(θ,ϕ): 5/2,7/2,5/2, dq W(θ,ϕ): 5/2,7/2,5/2, qq 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 emission angle θ/π angular correlation W (θ ,ϕ )

Figure 2-4: Angular correlation functions 𝑊 (𝜃, 𝜑) vs 𝜃 for unpolarized beams. Top: 𝑊 (𝜃, 𝜑) for three transitions in 238_{U. Bottom: 𝑊 (𝜃, 𝜑) for three transitions in}27_{Al. In both plots,}

the horizontal gridline at 𝑊 (𝜃, 𝜑) = 1 denotes isotropic scattering, while the vertical gridline denotes the angle of 𝜃𝑑= 125∘ used in the experiments of Chapter 4.

inducing NRF interactions; however, because the NRF process effectively takes forward-going photons and scatters them into 4𝜋 (according to 𝑊 (𝜃, 𝜑)), the forward-forward-going flux near the resonance energies 𝐸𝑟 will be preferentially attenuated (or ‘notched’—see Fig. 2-6)

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of NRF emission from the foil. The presence of the isotope of interest in the measurement object thus modifies the NRF signal from the baseline, producing a measurable change in signal determined by the isotopics and geometry of the measurement object.

𝜑0(𝐸) 𝜑𝑡(𝐸) 𝑑2𝑛𝑒 𝑑𝐸′𝑑Ω detector filter 𝐷 object 𝑋 foil 𝑥 𝑑𝑥 𝜋 − 𝜃 𝜃 +^𝑥 +^𝑧 +^𝑦

Figure 2-5: One-dimensional, single-isotope schematic for the NRF radiation transport model used to derive the expected NRF count rate. In a warhead verification scenario, the object would be a isotope) warhead to be measured, and the foil would be a (multi-isotope) encryption foil whose detailed construction (i.e., multiple layers of thicknesses 𝑋𝑘)

would be kept secret from the inspector.

E flux ϕ0(E) E flux ϕt(E) E' flux n(E')

Figure 2-6: Sketches of the flux energy spectrum throughout the NRF measurement—see Fig. 2-5. Axes not to scale. Left: the incident bremsstrahlung flux, 𝜑0(𝐸), extending from

zero to the electron beam energy 𝐸max. Center: the transmitted flux, 𝜑𝑡(𝐸), with notches

at the resonance energies 𝐸𝑟 marked by dashed lines. Right: the detected flux, 𝑛(𝐸′). NRF

peaks appear at the resonance energies and at energies corresponding to branched decays through an intermediate state. In reality, the notches are only Δ ∼ 1 eV wide while the peak widths are ∼1 keV primarily due to detector resolution.

To develop this picture quantitatively, in this section we build a semi-analytical model that takes an incident photon flux 𝜑0(𝐸)plus a set of geometry and cross section parameters,

solves the radiation transport problem in one dimension for the spectrum of photons emitted from the foil, and then considers the effects of realistic detectors. For simplicity, we will begin with the one-dimensional, single isotope, single resonance picture outlined in Fig. 2-5, and only examine the so-called ‘once-collided’ flux of NRF photons that traverse the warhead

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without interacting then undergo a single NRF interaction in the foil.

2.2.1 Once-collided NRF flux

The incident flux of photons 𝜑0(𝐸) can in theory be generated through a variety of means,

but the two most relevant mechanisms here are electron bremsstrahlung and laser Compton scattering (LCS). The former technique accelerates MeV-scale electrons into a metal target, producing a broad, continuous photon energy spectrum up to an energy endpoint given by the electron beam kinetic energy 𝐸max. The latter technique collides ∼100–1000 MeV

electrons with ∼1000 nm laser photons, producing quasi-monoenergetic (at a given scattering angle 𝜃) MeV-scale photons. This incident flux first interacts with the measurement object (a proxy warhead, in the experiments of Chapter 4) of thickness 𝐷. The photons undergo both resonant (NRF) and non-resonant (Compton scattering, pair production, and photoelectric absorption, collectively denoted ‘nr’) interactions, giving a transmitted flux according to simple exponential attenuation of

𝜑𝑡(𝐸) = 𝜑0(𝐸) exp(−𝐷[𝜇NRF(𝐸) + 𝜇nr(𝐸)]), (2.34)

where 𝜇 ≡ 𝑁𝜎 denotes the linear attenuation coefficient corresponding to a cross section 𝜎 and target number density 𝑁. Following Section 2.1.1, 𝜎NRF

𝑟 = 𝜇NRF/𝑁 is given by Eq. 2.10

or possibly by the approximations of Eq. 2.15 or 2.18. The number density can further be broken down as

𝑁 ≡ 𝑓 𝜌𝑁𝐴

𝐴 (2.35)

where 𝑁𝐴is Avogadro’s number, 𝐴 is the molar mass of the target material, and 0 ≤ 𝑓 ≤ 1

denotes the fraction of the target material capable of undergoing the particular interaction. For non-resonant interactions, 𝑓 = 1, while for NRF, 𝑓 is the mole fraction of the isotope of interest. The 𝜇NRF term in Eq. 2.34 provides sharp attenuation over a narrow range

of energies, creating a ‘notch’ in the forward-going energy spectrum. It should be noted again that Eq. 2.34 only considers the uncollided flux, and thus does not account for build-up, which, in the context of NRF, is called ‘notch refill,’ since higher-energy photons can downscatter into the notch and become on-resonance—see Section 2.2.2.

Conversely, in the foil, we are only interested in the collided flux that undergoes NRF and is emitted towards the detector. Because an NRF interaction can produce photons of multiple different energies according to the isotope’s nuclear level structure, we now make a distinction in notation between the incident photon energy 𝐸 and the emitted photon energy 𝐸′_{. The emission rate will thus depend on the branching ratio 𝑏}

𝑟,𝑗, which denotes

the fraction of decays that emit a photon of energy 𝐸′ _{= 𝐸 − 𝐸}

𝑗. Note here again we are

ignoring recoil.

In a slight shift of notation for later convenience, we will write the energy spectrum of photons emitted from the foil as a differential, 𝑑𝑛𝑒/𝑑𝐸′, instead of a simple function as

we did for 𝜑0(𝐸) and 𝜑𝑡(𝐸). We are also interested in the solid angle dependence of this

spectrum, 𝑑2_𝑛

𝑒/𝑑𝐸′𝑑Ω, and how the production rate of emitted NRF photons varies as the

available flux changes over the thickness of the foil,11 _𝑑3_𝑛

𝑒/𝑑𝐸′𝑑Ω𝑑𝑥. This triple-differential

rate of NRF photon emission in the energy range [𝐸′_{, 𝐸}′_{+ 𝑑𝐸}′_]_{into a solid angle element} 11_{For historical reasons, the infinitesimal foil thickness is written as 𝑑𝑥 rather than the 𝑑𝑧 that would}

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𝑑Ω due to NRF interactions in the foil element 𝑑𝑥 can be determined almost by inspection: it should be proportional to the available (transmitted) flux 𝜑𝑡(𝐸), the branching ratio 𝑏𝑟,𝑗,

and, since we are considering the flux emitted into the solid angle element 𝑑Ω, Eq. 2.36 must also be proportional to Eq. 2.28. Finally, we must introduce a correction factor to account for the attenuation of the flux within the foil itself, which we will denote for now as 𝑒−𝑎(𝑥,𝜃,𝐸,𝐸′). Putting everything together, we have12

𝑑3_𝑛 𝑒 𝑑𝐸′𝑑Ω 𝑑𝑥 = 𝜑𝑡(𝐸) 𝑏𝑟,𝑗 𝑑𝜇NRF(𝐸) 𝑑Ω 𝑒 −𝑎(𝑥,𝜃,𝐸,𝐸′₎ . (2.36)

For the moment, we will allow both 𝐸 and 𝐸′ _{to be present on the right-hand side of}

Eq. 2.36, despite only having a differential in 𝐸′_{. The correction factor 𝑒}−𝑎 _{can be found by}

considering the attenuation of photons traveling into and out of the foil. At a depth 𝑥 into the foil, the incoming flux available for NRF production will have been exponentially attenuated by both resonant and non-resonant processes in the preceding depth 𝑥, and any outgoing NRF flux must traverse a path length of 𝑥/ cos(𝜋 − 𝜃) to escape the foil. The outgoing NRF flux, however, is only subject to non-resonant attenuation, as recoil energy losses push the outgoing photon energy 𝐸′ _{far enough below resonance to ignore the 𝜇}

NRF(𝐸′) term. Thus we can write 𝑎(𝑥, 𝜃, 𝐸, 𝐸′) ≡ 𝑎in(𝑥, 𝐸) + 𝑎out(𝑥, 𝜃, 𝐸′) (2.37) = [𝜇NRF(𝐸) + 𝜇nr(𝐸)] 𝑥 +[︀𝜇nr(𝐸′) ]︀ 𝑥 cos(𝜋 − 𝜃). (2.38) The triple-differential emission rate becomes

𝑑3𝑛𝑒 𝑑𝐸′_{𝑑Ω 𝑑𝑥} = 𝜑𝑡(𝐸) 𝑏𝑟,𝑗𝜇NRF(𝐸) 𝑊 (𝜃, 𝜑) 4𝜋 × exp {︂ −𝑥 [︂ 𝜇NRF(𝐸) + 𝜇nr(𝐸) + 𝜇nr(𝐸′) cos(𝜋 − 𝜃) ]︂}︂ . (2.39) For concision, we group the various 𝜇 terms from the foil as

𝜇eff(𝐸, 𝐸′, 𝜃) ≡ 𝜇NRF(𝐸) + 𝜇nr(𝐸) +

𝜇nr(𝐸′)

cos(𝜋 − 𝜃), (2.40)

and integrate over the thickness of the foil from 𝑥 = 0 to 𝑥 = 𝑋 to obtain 𝑑2𝑛𝑒 𝑑𝐸′𝑑Ω = 𝜑𝑡(𝐸) 𝑏𝑟,𝑗𝜇NRF(𝐸) 𝑊 (𝜃, 𝜑) 4𝜋 1 − exp [−𝑋𝜇eff(𝐸, 𝐸′, 𝜃)] 𝜇eff(𝐸, 𝐸′, 𝜃) . (2.41)

Rigorously speaking, this is the answer to our radiation transport problem of finding the 12_{Here we assume a single-step cascade, so that the observed photon energy only arises from one transition.}

If the range of energies 𝐸 in the incident flux 𝜑0(𝐸)extends higher than the NRF resonance energies 𝐸𝑟 of

interest, the resonant levels can be populated not just via excitation from the ground state but also by the decay from higher-energy excited states. In this ‘feeding’ process, more complicated cascades and 𝑊 (𝜃, 𝜑) functions can arise, and a careful sum over Eq. 2.36 must be computed. The 2.212 MeV level of 27_{Al is a}

notable example: it may be reached as an excitation from the ground state or as an intermediate state from the decay of at least 46 different states with level energy 𝐸𝑟> 2.7MeV [44]. In this work, the photon beam

energies are ≤2.7 MeV, so the strength of the 2.212 MeV line is driven entirely by the transition from the ground state to the 2.212 MeV level followed by decay back to the ground state.

Development of an isotope-sensitive warhead verification technique using nuclear resonance fluorescence